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<title>Chapter 1 Background mathematics | Fundamental statistical concepts and techniques in the biological and environmental sciences: With jamovi</title>
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<ul class="summary">
<li><a href="./">Statistics with jamovi</a></li>
<li class="divider"></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html"><i class="fa fa-check"></i>Preface</a>
<ul>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#structure"><i class="fa fa-check"></i>How this book is structured</a></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#datasets"><i class="fa fa-check"></i>Datasets used in this book</a></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#acknowledgements"><i class="fa fa-check"></i>Acknowledgements</a></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#author"><i class="fa fa-check"></i>About the author</a></li>
</ul></li>
<li class="chapter" data-level="1" data-path="Chapter_1.html"><a href="Chapter_1.html"><i class="fa fa-check"></i><b>1</b> Background mathematics</a>
<ul>
<li class="chapter" data-level="1.1" data-path="Chapter_1.html"><a href="Chapter_1.html#numbers-and-operations"><i class="fa fa-check"></i><b>1.1</b> Numbers and operations</a></li>
<li class="chapter" data-level="1.2" data-path="Chapter_1.html"><a href="Chapter_1.html#logarithms"><i class="fa fa-check"></i><b>1.2</b> Logarithms</a></li>
<li class="chapter" data-level="1.3" data-path="Chapter_1.html"><a href="Chapter_1.html#order-of-operations"><i class="fa fa-check"></i><b>1.3</b> Order of operations</a></li>
</ul></li>
<li class="chapter" data-level="2" data-path="Chapter_2.html"><a href="Chapter_2.html"><i class="fa fa-check"></i><b>2</b> Data organisation</a>
<ul>
<li class="chapter" data-level="2.1" data-path="Chapter_2.html"><a href="Chapter_2.html#tidy-data"><i class="fa fa-check"></i><b>2.1</b> Tidy data</a></li>
<li class="chapter" data-level="2.2" data-path="Chapter_2.html"><a href="Chapter_2.html#data-files"><i class="fa fa-check"></i><b>2.2</b> Data files</a></li>
<li class="chapter" data-level="2.3" data-path="Chapter_2.html"><a href="Chapter_2.html#managing-data-files"><i class="fa fa-check"></i><b>2.3</b> Managing data files</a></li>
</ul></li>
<li class="chapter" data-level="3" data-path="Chapter_3.html"><a href="Chapter_3.html"><i class="fa fa-check"></i><b>3</b> <em>Practical</em>. Preparing data</a>
<ul>
<li class="chapter" data-level="3.1" data-path="Chapter_3.html"><a href="Chapter_3.html#transferring-data-to-a-spreadsheet"><i class="fa fa-check"></i><b>3.1</b> Transferring data to a spreadsheet</a></li>
<li class="chapter" data-level="3.2" data-path="Chapter_3.html"><a href="Chapter_3.html#making-spreadsheet-data-tidy"><i class="fa fa-check"></i><b>3.2</b> Making spreadsheet data tidy</a></li>
<li class="chapter" data-level="3.3" data-path="Chapter_3.html"><a href="Chapter_3.html#making-data-tidy-again"><i class="fa fa-check"></i><b>3.3</b> Making data tidy again</a></li>
<li class="chapter" data-level="3.4" data-path="Chapter_3.html"><a href="Chapter_3.html#tidy-data-and-spreadsheet-calculations"><i class="fa fa-check"></i><b>3.4</b> Tidy data and spreadsheet calculations</a></li>
<li class="chapter" data-level="3.5" data-path="Chapter_3.html"><a href="Chapter_3.html#summary"><i class="fa fa-check"></i><b>3.5</b> Summary</a></li>
</ul></li>
<li class="chapter" data-level="4" data-path="Chapter_4.html"><a href="Chapter_4.html"><i class="fa fa-check"></i><b>4</b> Populations and samples</a></li>
<li class="chapter" data-level="5" data-path="Chapter_5.html"><a href="Chapter_5.html"><i class="fa fa-check"></i><b>5</b> Types of variables</a></li>
<li class="chapter" data-level="6" data-path="Chapter_6.html"><a href="Chapter_6.html"><i class="fa fa-check"></i><b>6</b> Accuracy, precision, and units</a>
<ul>
<li class="chapter" data-level="6.1" data-path="Chapter_6.html"><a href="Chapter_6.html#accuracy"><i class="fa fa-check"></i><b>6.1</b> Accuracy</a></li>
<li class="chapter" data-level="6.2" data-path="Chapter_6.html"><a href="Chapter_6.html#precision"><i class="fa fa-check"></i><b>6.2</b> Precision</a></li>
<li class="chapter" data-level="6.3" data-path="Chapter_6.html"><a href="Chapter_6.html#systems-of-units"><i class="fa fa-check"></i><b>6.3</b> Systems of units</a></li>
</ul></li>
<li class="chapter" data-level="7" data-path="Chapter_7.html"><a href="Chapter_7.html"><i class="fa fa-check"></i><b>7</b> Uncertainty propagation</a>
<ul>
<li class="chapter" data-level="7.1" data-path="Chapter_7.html"><a href="Chapter_7.html#adding-or-subtracting-errors"><i class="fa fa-check"></i><b>7.1</b> Adding or subtracting errors</a></li>
<li class="chapter" data-level="7.2" data-path="Chapter_7.html"><a href="Chapter_7.html#multiplying-or-dividing-errors"><i class="fa fa-check"></i><b>7.2</b> Multiplying or dividing errors</a></li>
</ul></li>
<li class="chapter" data-level="8" data-path="Chapter_8.html"><a href="Chapter_8.html"><i class="fa fa-check"></i><b>8</b> <em>Practical</em>. Introduction to jamovi</a>
<ul>
<li class="chapter" data-level="8.1" data-path="Chapter_8.html"><a href="Chapter_8.html#summary_statistics_02"><i class="fa fa-check"></i><b>8.1</b> Summary statistics</a></li>
<li class="chapter" data-level="8.2" data-path="Chapter_8.html"><a href="Chapter_8.html#transforming_variables_02"><i class="fa fa-check"></i><b>8.2</b> Transforming variables</a></li>
<li class="chapter" data-level="8.3" data-path="Chapter_8.html"><a href="Chapter_8.html#computing_variables_02"><i class="fa fa-check"></i><b>8.3</b> Computing variables</a></li>
<li class="chapter" data-level="8.4" data-path="Chapter_8.html"><a href="Chapter_8.html#summary-1"><i class="fa fa-check"></i><b>8.4</b> Summary</a></li>
</ul></li>
<li class="chapter" data-level="9" data-path="Chapter_9.html"><a href="Chapter_9.html"><i class="fa fa-check"></i><b>9</b> Decimal places, significant figures, and rounding</a>
<ul>
<li class="chapter" data-level="9.1" data-path="Chapter_9.html"><a href="Chapter_9.html#decimal-places-and-significant-figures"><i class="fa fa-check"></i><b>9.1</b> Decimal places and significant figures</a></li>
<li class="chapter" data-level="9.2" data-path="Chapter_9.html"><a href="Chapter_9.html#rounding"><i class="fa fa-check"></i><b>9.2</b> Rounding</a></li>
</ul></li>
<li class="chapter" data-level="10" data-path="Chapter_10.html"><a href="Chapter_10.html"><i class="fa fa-check"></i><b>10</b> Graphs</a>
<ul>
<li class="chapter" data-level="10.1" data-path="Chapter_10.html"><a href="Chapter_10.html#histograms"><i class="fa fa-check"></i><b>10.1</b> Histograms</a></li>
<li class="chapter" data-level="10.2" data-path="Chapter_10.html"><a href="Chapter_10.html#barplots-and-pie-charts"><i class="fa fa-check"></i><b>10.2</b> Barplots and pie charts</a></li>
<li class="chapter" data-level="10.3" data-path="Chapter_10.html"><a href="Chapter_10.html#box-whisker-plots"><i class="fa fa-check"></i><b>10.3</b> Box-whisker plots</a></li>
</ul></li>
<li class="chapter" data-level="11" data-path="Chapter_11.html"><a href="Chapter_11.html"><i class="fa fa-check"></i><b>11</b> Measures of central tendency</a>
<ul>
<li class="chapter" data-level="11.1" data-path="Chapter_11.html"><a href="Chapter_11.html#the-mean"><i class="fa fa-check"></i><b>11.1</b> The mean</a></li>
<li class="chapter" data-level="11.2" data-path="Chapter_11.html"><a href="Chapter_11.html#the-mode"><i class="fa fa-check"></i><b>11.2</b> The mode</a></li>
<li class="chapter" data-level="11.3" data-path="Chapter_11.html"><a href="Chapter_11.html#the-median-and-quantiles"><i class="fa fa-check"></i><b>11.3</b> The median and quantiles</a></li>
</ul></li>
<li class="chapter" data-level="12" data-path="Chapter_12.html"><a href="Chapter_12.html"><i class="fa fa-check"></i><b>12</b> Measures of spread</a>
<ul>
<li class="chapter" data-level="12.1" data-path="Chapter_12.html"><a href="Chapter_12.html#the-range"><i class="fa fa-check"></i><b>12.1</b> The range</a></li>
<li class="chapter" data-level="12.2" data-path="Chapter_12.html"><a href="Chapter_12.html#the-inter-quartile-range"><i class="fa fa-check"></i><b>12.2</b> The inter-quartile range</a></li>
<li class="chapter" data-level="12.3" data-path="Chapter_12.html"><a href="Chapter_12.html#the-variance"><i class="fa fa-check"></i><b>12.3</b> The variance</a></li>
<li class="chapter" data-level="12.4" data-path="Chapter_12.html"><a href="Chapter_12.html#the-standard-deviation"><i class="fa fa-check"></i><b>12.4</b> The standard deviation</a></li>
<li class="chapter" data-level="12.5" data-path="Chapter_12.html"><a href="Chapter_12.html#the-coefficient-of-variation"><i class="fa fa-check"></i><b>12.5</b> The coefficient of variation</a></li>
<li class="chapter" data-level="12.6" data-path="Chapter_12.html"><a href="Chapter_12.html#the-standard-error"><i class="fa fa-check"></i><b>12.6</b> The standard error</a></li>
</ul></li>
<li class="chapter" data-level="13" data-path="Chapter_13.html"><a href="Chapter_13.html"><i class="fa fa-check"></i><b>13</b> Skew and kurtosis</a>
<ul>
<li class="chapter" data-level="13.1" data-path="Chapter_13.html"><a href="Chapter_13.html#skew"><i class="fa fa-check"></i><b>13.1</b> Skew</a></li>
<li class="chapter" data-level="13.2" data-path="Chapter_13.html"><a href="Chapter_13.html#kurtosis"><i class="fa fa-check"></i><b>13.2</b> Kurtosis</a></li>
<li class="chapter" data-level="13.3" data-path="Chapter_13.html"><a href="Chapter_13.html#moments"><i class="fa fa-check"></i><b>13.3</b> Moments</a></li>
</ul></li>
<li class="chapter" data-level="14" data-path="Chapter_14.html"><a href="Chapter_14.html"><i class="fa fa-check"></i><b>14</b> <em>Practical</em>. Plotting and statistical summaries in jamovi</a>
<ul>
<li class="chapter" data-level="14.1" data-path="Chapter_14.html"><a href="Chapter_14.html#reorganise-the-dataset-into-a-tidy-format"><i class="fa fa-check"></i><b>14.1</b> Reorganise the dataset into a tidy format</a></li>
<li class="chapter" data-level="14.2" data-path="Chapter_14.html"><a href="Chapter_14.html#histograms-and-box-whisker-plots"><i class="fa fa-check"></i><b>14.2</b> Histograms and box-whisker plots</a></li>
<li class="chapter" data-level="14.3" data-path="Chapter_14.html"><a href="Chapter_14.html#calculate-summary-statistics"><i class="fa fa-check"></i><b>14.3</b> Calculate summary statistics</a></li>
<li class="chapter" data-level="14.4" data-path="Chapter_14.html"><a href="Chapter_14.html#reporting-decimals-and-significant-figures"><i class="fa fa-check"></i><b>14.4</b> Reporting decimals and significant figures</a></li>
<li class="chapter" data-level="14.5" data-path="Chapter_14.html"><a href="Chapter_14.html#comparing-across-sites"><i class="fa fa-check"></i><b>14.5</b> Comparing across sites</a></li>
</ul></li>
<li class="chapter" data-level="15" data-path="Chapter_15.html"><a href="Chapter_15.html"><i class="fa fa-check"></i><b>15</b> Introduction to probability models</a>
<ul>
<li class="chapter" data-level="15.1" data-path="Chapter_15.html"><a href="Chapter_15.html#instructive-example"><i class="fa fa-check"></i><b>15.1</b> Instructive example</a></li>
<li class="chapter" data-level="15.2" data-path="Chapter_15.html"><a href="Chapter_15.html#biological-applications"><i class="fa fa-check"></i><b>15.2</b> Biological applications</a></li>
<li class="chapter" data-level="15.3" data-path="Chapter_15.html"><a href="Chapter_15.html#sampling-with-and-without-replacement"><i class="fa fa-check"></i><b>15.3</b> Sampling with and without replacement</a></li>
<li class="chapter" data-level="15.4" data-path="Chapter_15.html"><a href="Chapter_15.html#probability-distributions"><i class="fa fa-check"></i><b>15.4</b> Probability distributions</a>
<ul>
<li class="chapter" data-level="15.4.1" data-path="Chapter_15.html"><a href="Chapter_15.html#binomial-distribution"><i class="fa fa-check"></i><b>15.4.1</b> Binomial distribution</a></li>
<li class="chapter" data-level="15.4.2" data-path="Chapter_15.html"><a href="Chapter_15.html#poisson-distribution"><i class="fa fa-check"></i><b>15.4.2</b> Poisson distribution</a></li>
<li class="chapter" data-level="15.4.3" data-path="Chapter_15.html"><a href="Chapter_15.html#uniform-distribution"><i class="fa fa-check"></i><b>15.4.3</b> Uniform distribution</a></li>
<li class="chapter" data-level="15.4.4" data-path="Chapter_15.html"><a href="Chapter_15.html#normal-distribution"><i class="fa fa-check"></i><b>15.4.4</b> Normal distribution</a></li>
</ul></li>
<li class="chapter" data-level="15.5" data-path="Chapter_15.html"><a href="Chapter_15.html#summary-2"><i class="fa fa-check"></i><b>15.5</b> Summary</a></li>
</ul></li>
<li class="chapter" data-level="16" data-path="Chapter_16.html"><a href="Chapter_16.html"><i class="fa fa-check"></i><b>16</b> Central Limit Theorem</a>
<ul>
<li class="chapter" data-level="16.1" data-path="Chapter_16.html"><a href="Chapter_16.html#the-distribution-of-means-is-normal"><i class="fa fa-check"></i><b>16.1</b> The distribution of means is normal</a></li>
<li class="chapter" data-level="16.2" data-path="Chapter_16.html"><a href="Chapter_16.html#probability-and-z-scores"><i class="fa fa-check"></i><b>16.2</b> Probability and z-scores</a></li>
</ul></li>
<li class="chapter" data-level="17" data-path="Chapter_17.html"><a href="Chapter_17.html"><i class="fa fa-check"></i><b>17</b> <em>Practical</em>. Probability and simulation</a>
<ul>
<li class="chapter" data-level="17.1" data-path="Chapter_17.html"><a href="Chapter_17.html#probabilities-from-a-dataset"><i class="fa fa-check"></i><b>17.1</b> Probabilities from a dataset</a></li>
<li class="chapter" data-level="17.2" data-path="Chapter_17.html"><a href="Chapter_17.html#probabilities-from-a-normal-distribution"><i class="fa fa-check"></i><b>17.2</b> Probabilities from a normal distribution</a></li>
<li class="chapter" data-level="17.3" data-path="Chapter_17.html"><a href="Chapter_17.html#central-limit-theorem"><i class="fa fa-check"></i><b>17.3</b> Central limit theorem</a></li>
</ul></li>
<li class="chapter" data-level="18" data-path="Chapter_18.html"><a href="Chapter_18.html"><i class="fa fa-check"></i><b>18</b> Confidence intervals</a>
<ul>
<li class="chapter" data-level="18.1" data-path="Chapter_18.html"><a href="Chapter_18.html#normal-distribution-cis"><i class="fa fa-check"></i><b>18.1</b> Normal distribution CIs</a></li>
<li class="chapter" data-level="18.2" data-path="Chapter_18.html"><a href="Chapter_18.html#binomial-distribution-cis"><i class="fa fa-check"></i><b>18.2</b> Binomial distribution CIs</a></li>
</ul></li>
<li class="chapter" data-level="19" data-path="Chapter_19.html"><a href="Chapter_19.html"><i class="fa fa-check"></i><b>19</b> The t-interval</a></li>
<li class="chapter" data-level="20" data-path="Chapter_20.html"><a href="Chapter_20.html"><i class="fa fa-check"></i><b>20</b> <em>Practical</em>. z- and t-intervals</a>
<ul>
<li class="chapter" data-level="20.1" data-path="Chapter_20.html"><a href="Chapter_20.html#confidence-intervals-with-distraction"><i class="fa fa-check"></i><b>20.1</b> Confidence intervals with distrACTION</a></li>
<li class="chapter" data-level="20.2" data-path="Chapter_20.html"><a href="Chapter_20.html#confidence-intervals-from-z--and-t-scores"><i class="fa fa-check"></i><b>20.2</b> Confidence intervals from z- and t-scores</a></li>
<li class="chapter" data-level="20.3" data-path="Chapter_20.html"><a href="Chapter_20.html#confidence-intervals-for-different-sample-sizes"><i class="fa fa-check"></i><b>20.3</b> Confidence intervals for different sample sizes</a></li>
<li class="chapter" data-level="20.4" data-path="Chapter_20.html"><a href="Chapter_20.html#proportion-confidence-intervals"><i class="fa fa-check"></i><b>20.4</b> Proportion confidence intervals</a></li>
<li class="chapter" data-level="20.5" data-path="Chapter_20.html"><a href="Chapter_20.html#another-proportion-confidence-interval"><i class="fa fa-check"></i><b>20.5</b> Another proportion confidence interval</a></li>
</ul></li>
<li class="chapter" data-level="21" data-path="Chapter_21.html"><a href="Chapter_21.html"><i class="fa fa-check"></i><b>21</b> What is hypothesis testing?</a>
<ul>
<li class="chapter" data-level="21.1" data-path="Chapter_21.html"><a href="Chapter_21.html#how-ridiculous-is-our-hypothesis"><i class="fa fa-check"></i><b>21.1</b> How ridiculous is our hypothesis?</a></li>
<li class="chapter" data-level="21.2" data-path="Chapter_21.html"><a href="Chapter_21.html#statistical-hypothesis-testing"><i class="fa fa-check"></i><b>21.2</b> Statistical hypothesis testing</a></li>
<li class="chapter" data-level="21.3" data-path="Chapter_21.html"><a href="Chapter_21.html#p-values-false-positives-and-power"><i class="fa fa-check"></i><b>21.3</b> P-values, false positives, and power</a></li>
</ul></li>
<li class="chapter" data-level="22" data-path="Chapter_22.html"><a href="Chapter_22.html"><i class="fa fa-check"></i><b>22</b> The t-test</a>
<ul>
<li class="chapter" data-level="22.1" data-path="Chapter_22.html"><a href="Chapter_22.html#one-sample-t-test"><i class="fa fa-check"></i><b>22.1</b> One sample t-test</a></li>
<li class="chapter" data-level="22.2" data-path="Chapter_22.html"><a href="Chapter_22.html#independent-samples-t-test"><i class="fa fa-check"></i><b>22.2</b> Independent samples t-test</a></li>
<li class="chapter" data-level="22.3" data-path="Chapter_22.html"><a href="Chapter_22.html#paired-samples-t-test"><i class="fa fa-check"></i><b>22.3</b> Paired samples t-test</a></li>
<li class="chapter" data-level="22.4" data-path="Chapter_22.html"><a href="Chapter_22.html#assumptions-of-t-tests"><i class="fa fa-check"></i><b>22.4</b> Assumptions of t-tests</a></li>
<li class="chapter" data-level="22.5" data-path="Chapter_22.html"><a href="Chapter_22.html#non-parametric-alternatives"><i class="fa fa-check"></i><b>22.5</b> Non-parametric alternatives</a>
<ul>
<li class="chapter" data-level="22.5.1" data-path="Chapter_22.html"><a href="Chapter_22.html#wilcoxon-test"><i class="fa fa-check"></i><b>22.5.1</b> Wilcoxon test</a></li>
<li class="chapter" data-level="22.5.2" data-path="Chapter_22.html"><a href="Chapter_22.html#mann-whitney-u-test"><i class="fa fa-check"></i><b>22.5.2</b> Mann-Whitney U test</a></li>
</ul></li>
<li class="chapter" data-level="22.6" data-path="Chapter_22.html"><a href="Chapter_22.html#summary-3"><i class="fa fa-check"></i><b>22.6</b> Summary</a></li>
</ul></li>
<li class="chapter" data-level="23" data-path="Chapter_23.html"><a href="Chapter_23.html"><i class="fa fa-check"></i><b>23</b> <em>Practical</em>. Hypothesis testing and t-tests</a>
<ul>
<li class="chapter" data-level="23.1" data-path="Chapter_23.html"><a href="Chapter_23.html#one-sample-t-test-1"><i class="fa fa-check"></i><b>23.1</b> One sample t-test</a></li>
<li class="chapter" data-level="23.2" data-path="Chapter_23.html"><a href="Chapter_23.html#paired-t-test"><i class="fa fa-check"></i><b>23.2</b> Paired t-test</a></li>
<li class="chapter" data-level="23.3" data-path="Chapter_23.html"><a href="Chapter_23.html#wilcoxon-test-1"><i class="fa fa-check"></i><b>23.3</b> Wilcoxon test</a></li>
<li class="chapter" data-level="23.4" data-path="Chapter_23.html"><a href="Chapter_23.html#independent-samples-t-test-1"><i class="fa fa-check"></i><b>23.4</b> Independent samples t-test</a></li>
<li class="chapter" data-level="23.5" data-path="Chapter_23.html"><a href="Chapter_23.html#mann-whitney-u-test-1"><i class="fa fa-check"></i><b>23.5</b> Mann-Whitney U Test</a></li>
</ul></li>
<li class="chapter" data-level="24" data-path="Chapter_24.html"><a href="Chapter_24.html"><i class="fa fa-check"></i><b>24</b> Analysis of variance</a>
<ul>
<li class="chapter" data-level="24.1" data-path="Chapter_24.html"><a href="Chapter_24.html#f-distribution"><i class="fa fa-check"></i><b>24.1</b> F-distribution</a></li>
<li class="chapter" data-level="24.2" data-path="Chapter_24.html"><a href="Chapter_24.html#one-way-anova"><i class="fa fa-check"></i><b>24.2</b> One-way ANOVA</a>
<ul>
<li class="chapter" data-level="24.2.1" data-path="Chapter_24.html"><a href="Chapter_24.html#anova-mean-variance-among-groups"><i class="fa fa-check"></i><b>24.2.1</b> ANOVA mean variance among groups</a></li>
<li class="chapter" data-level="24.2.2" data-path="Chapter_24.html"><a href="Chapter_24.html#anova-mean-variance-within-groups"><i class="fa fa-check"></i><b>24.2.2</b> ANOVA mean variance within groups</a></li>
<li class="chapter" data-level="24.2.3" data-path="Chapter_24.html"><a href="Chapter_24.html#anova-f-statistic-calculation"><i class="fa fa-check"></i><b>24.2.3</b> ANOVA F-statistic calculation</a></li>
</ul></li>
<li class="chapter" data-level="24.3" data-path="Chapter_24.html"><a href="Chapter_24.html#assumptions-of-anova"><i class="fa fa-check"></i><b>24.3</b> Assumptions of ANOVA</a></li>
</ul></li>
<li class="chapter" data-level="25" data-path="Chapter_25.html"><a href="Chapter_25.html"><i class="fa fa-check"></i><b>25</b> Multiple comparisons</a></li>
<li class="chapter" data-level="26" data-path="Chapter_26.html"><a href="Chapter_26.html"><i class="fa fa-check"></i><b>26</b> Kruskal-Wallis H test</a></li>
<li class="chapter" data-level="27" data-path="Chapter_27.html"><a href="Chapter_27.html"><i class="fa fa-check"></i><b>27</b> Two-way ANOVA</a></li>
<li class="chapter" data-level="28" data-path="Chapter_28.html"><a href="Chapter_28.html"><i class="fa fa-check"></i><b>28</b> <em>Practical</em>. ANOVA and associated tests</a>
<ul>
<li class="chapter" data-level="28.1" data-path="Chapter_28.html"><a href="Chapter_28.html#one-way-anova-site"><i class="fa fa-check"></i><b>28.1</b> One-way ANOVA (site)</a></li>
<li class="chapter" data-level="28.2" data-path="Chapter_28.html"><a href="Chapter_28.html#one-way-anova-profile"><i class="fa fa-check"></i><b>28.2</b> One-way ANOVA (profile)</a></li>
<li class="chapter" data-level="28.3" data-path="Chapter_28.html"><a href="Chapter_28.html#multiple-comparisons"><i class="fa fa-check"></i><b>28.3</b> Multiple comparisons</a></li>
<li class="chapter" data-level="28.4" data-path="Chapter_28.html"><a href="Chapter_28.html#kruskal-wallis-h-test"><i class="fa fa-check"></i><b>28.4</b> Kruskal-Wallis H test</a></li>
<li class="chapter" data-level="28.5" data-path="Chapter_28.html"><a href="Chapter_28.html#two-way-anova"><i class="fa fa-check"></i><b>28.5</b> Two-way ANOVA</a></li>
</ul></li>
<li class="chapter" data-level="29" data-path="Chapter_29.html"><a href="Chapter_29.html"><i class="fa fa-check"></i><b>29</b> Frequency and count data</a>
<ul>
<li class="chapter" data-level="29.1" data-path="Chapter_29.html"><a href="Chapter_29.html#chi-square-distribution"><i class="fa fa-check"></i><b>29.1</b> Chi-square distribution</a></li>
<li class="chapter" data-level="29.2" data-path="Chapter_29.html"><a href="Chapter_29.html#chi-square-goodness-of-fit"><i class="fa fa-check"></i><b>29.2</b> Chi-square goodness of fit</a></li>
<li class="chapter" data-level="29.3" data-path="Chapter_29.html"><a href="Chapter_29.html#chi-square-test-of-association"><i class="fa fa-check"></i><b>29.3</b> Chi-square test of association</a></li>
</ul></li>
<li class="chapter" data-level="30" data-path="Chapter_30.html"><a href="Chapter_30.html"><i class="fa fa-check"></i><b>30</b> Correlation</a>
<ul>
<li class="chapter" data-level="30.1" data-path="Chapter_30.html"><a href="Chapter_30.html#scatterplots"><i class="fa fa-check"></i><b>30.1</b> Scatterplots</a></li>
<li class="chapter" data-level="30.2" data-path="Chapter_30.html"><a href="Chapter_30.html#correlation-coefficient"><i class="fa fa-check"></i><b>30.2</b> Correlation coefficient</a>
<ul>
<li class="chapter" data-level="30.2.1" data-path="Chapter_30.html"><a href="Chapter_30.html#pearson-product-moment-correlation-coefficient"><i class="fa fa-check"></i><b>30.2.1</b> Pearson product moment correlation coefficient</a></li>
<li class="chapter" data-level="30.2.2" data-path="Chapter_30.html"><a href="Chapter_30.html#spearmans-rank-correlation-coefficient"><i class="fa fa-check"></i><b>30.2.2</b> Spearman’s rank correlation coefficient</a></li>
</ul></li>
<li class="chapter" data-level="30.3" data-path="Chapter_30.html"><a href="Chapter_30.html#correlation-hypothesis-testing"><i class="fa fa-check"></i><b>30.3</b> Correlation hypothesis testing</a></li>
</ul></li>
<li class="chapter" data-level="31" data-path="Chapter_31.html"><a href="Chapter_31.html"><i class="fa fa-check"></i><b>31</b> <em>Practical</em>. Analysis of counts and correlations</a>
<ul>
<li class="chapter" data-level="31.1" data-path="Chapter_31.html"><a href="Chapter_31.html#survival-goodness-of-fit"><i class="fa fa-check"></i><b>31.1</b> Survival goodness of fit</a></li>
<li class="chapter" data-level="31.2" data-path="Chapter_31.html"><a href="Chapter_31.html#colony-goodness-of-fit"><i class="fa fa-check"></i><b>31.2</b> Colony goodness of fit</a></li>
<li class="chapter" data-level="31.3" data-path="Chapter_31.html"><a href="Chapter_31.html#chi-square-test-of-association-1"><i class="fa fa-check"></i><b>31.3</b> Chi-Square test of association</a></li>
<li class="chapter" data-level="31.4" data-path="Chapter_31.html"><a href="Chapter_31.html#pearson-product-moment-correlation-test"><i class="fa fa-check"></i><b>31.4</b> Pearson product moment correlation test</a></li>
<li class="chapter" data-level="31.5" data-path="Chapter_31.html"><a href="Chapter_31.html#spearmans-rank-correlation-test"><i class="fa fa-check"></i><b>31.5</b> Spearman’s rank correlation test</a></li>
<li class="chapter" data-level="31.6" data-path="Chapter_31.html"><a href="Chapter_31.html#untidy-goodness-of-fit"><i class="fa fa-check"></i><b>31.6</b> Untidy goodness of fit</a></li>
</ul></li>
<li class="chapter" data-level="32" data-path="Chapter_32.html"><a href="Chapter_32.html"><i class="fa fa-check"></i><b>32</b> Simple linear regression</a>
<ul>
<li class="chapter" data-level="32.1" data-path="Chapter_32.html"><a href="Chapter_32.html#visual-interpretation-of-regression"><i class="fa fa-check"></i><b>32.1</b> Visual interpretation of regression</a></li>
<li class="chapter" data-level="32.2" data-path="Chapter_32.html"><a href="Chapter_32.html#intercepts-slopes-and-residuals"><i class="fa fa-check"></i><b>32.2</b> Intercepts, slopes, and residuals</a></li>
<li class="chapter" data-level="32.3" data-path="Chapter_32.html"><a href="Chapter_32.html#regression-coefficients"><i class="fa fa-check"></i><b>32.3</b> Regression coefficients</a></li>
<li class="chapter" data-level="32.4" data-path="Chapter_32.html"><a href="Chapter_32.html#regression-line-calculation"><i class="fa fa-check"></i><b>32.4</b> Regression line calculation</a></li>
<li class="chapter" data-level="32.5" data-path="Chapter_32.html"><a href="Chapter_32.html#coefficient-of-determination"><i class="fa fa-check"></i><b>32.5</b> Coefficient of determination</a></li>
<li class="chapter" data-level="32.6" data-path="Chapter_32.html"><a href="Chapter_32.html#regression-assumptions"><i class="fa fa-check"></i><b>32.6</b> Regression assumptions</a></li>
<li class="chapter" data-level="32.7" data-path="Chapter_32.html"><a href="Chapter_32.html#regression-hypothesis-testing"><i class="fa fa-check"></i><b>32.7</b> Regression hypothesis testing</a>
<ul>
<li class="chapter" data-level="32.7.1" data-path="Chapter_32.html"><a href="Chapter_32.html#overall-model-significance"><i class="fa fa-check"></i><b>32.7.1</b> Overall model significance</a></li>
<li class="chapter" data-level="32.7.2" data-path="Chapter_32.html"><a href="Chapter_32.html#significance-of-the-intercept"><i class="fa fa-check"></i><b>32.7.2</b> Significance of the intercept</a></li>
<li class="chapter" data-level="32.7.3" data-path="Chapter_32.html"><a href="Chapter_32.html#significance-of-the-slope"><i class="fa fa-check"></i><b>32.7.3</b> Significance of the slope</a></li>
<li class="chapter" data-level="32.7.4" data-path="Chapter_32.html"><a href="Chapter_32.html#simple-regression-output"><i class="fa fa-check"></i><b>32.7.4</b> Simple regression output</a></li>
</ul></li>
<li class="chapter" data-level="32.8" data-path="Chapter_32.html"><a href="Chapter_32.html#prediction-with-linear-models"><i class="fa fa-check"></i><b>32.8</b> Prediction with linear models</a></li>
<li class="chapter" data-level="32.9" data-path="Chapter_32.html"><a href="Chapter_32.html#conclusion"><i class="fa fa-check"></i><b>32.9</b> Conclusion</a></li>
</ul></li>
<li class="chapter" data-level="33" data-path="Chapter_33.html"><a href="Chapter_33.html"><i class="fa fa-check"></i><b>33</b> Multiple regression</a>
<ul>
<li class="chapter" data-level="33.1" data-path="Chapter_33.html"><a href="Chapter_33.html#adjusted-coefficient-of-determination"><i class="fa fa-check"></i><b>33.1</b> Adjusted coefficient of determination</a></li>
</ul></li>
<li class="chapter" data-level="34" data-path="Chapter_34.html"><a href="Chapter_34.html"><i class="fa fa-check"></i><b>34</b> <em>Practical</em>. Using regression</a>
<ul>
<li class="chapter" data-level="34.1" data-path="Chapter_34.html"><a href="Chapter_34.html#predicting-pyrogenic-carbon-from-soil-depth"><i class="fa fa-check"></i><b>34.1</b> Predicting pyrogenic carbon from soil depth</a></li>
<li class="chapter" data-level="34.2" data-path="Chapter_34.html"><a href="Chapter_34.html#predicting-pyrogenic-carbon-from-fire-frequency"><i class="fa fa-check"></i><b>34.2</b> Predicting pyrogenic carbon from fire frequency</a></li>
<li class="chapter" data-level="34.3" data-path="Chapter_34.html"><a href="Chapter_34.html#multiple-regression-depth-and-fire-frequency"><i class="fa fa-check"></i><b>34.3</b> Multiple regression depth and fire frequency</a></li>
<li class="chapter" data-level="34.4" data-path="Chapter_34.html"><a href="Chapter_34.html#large-multiple-regression"><i class="fa fa-check"></i><b>34.4</b> Large multiple regression</a></li>
<li class="chapter" data-level="34.5" data-path="Chapter_34.html"><a href="Chapter_34.html#predicting-temperature-from-fire-frequency"><i class="fa fa-check"></i><b>34.5</b> Predicting temperature from fire frequency</a></li>
</ul></li>
<li class="chapter" data-level="35" data-path="Chapter_35.html"><a href="Chapter_35.html"><i class="fa fa-check"></i><b>35</b> Randomisation</a>
<ul>
<li class="chapter" data-level="35.1" data-path="Chapter_35.html"><a href="Chapter_35.html#summary-of-parametric-hypothesis-testing"><i class="fa fa-check"></i><b>35.1</b> Summary of parametric hypothesis testing</a></li>
<li class="chapter" data-level="35.2" data-path="Chapter_35.html"><a href="Chapter_35.html#randomisation-approach"><i class="fa fa-check"></i><b>35.2</b> Randomisation approach</a></li>
<li class="chapter" data-level="35.3" data-path="Chapter_35.html"><a href="Chapter_35.html#randomisation-for-hypothesis-testing"><i class="fa fa-check"></i><b>35.3</b> Randomisation for hypothesis testing</a></li>
<li class="chapter" data-level="35.4" data-path="Chapter_35.html"><a href="Chapter_35.html#randomisation-assumptions"><i class="fa fa-check"></i><b>35.4</b> Randomisation assumptions</a></li>
<li class="chapter" data-level="35.5" data-path="Chapter_35.html"><a href="Chapter_35.html#bootstrapping"><i class="fa fa-check"></i><b>35.5</b> Bootstrapping</a></li>
<li class="chapter" data-level="35.6" data-path="Chapter_35.html"><a href="Chapter_35.html#randomisation-conclusions"><i class="fa fa-check"></i><b>35.6</b> Randomisation conclusions</a></li>
</ul></li>
<li class="appendix"><span><b>Appendix</b></span></li>
<li class="chapter" data-level="A" data-path="appendexA.html"><a href="appendexA.html"><i class="fa fa-check"></i><b>A</b> Answers to chapter exercises</a>
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<li class="chapter" data-level="A.1" data-path="appendexA.html"><a href="appendexA.html#chapter-3"><i class="fa fa-check"></i><b>A.1</b> Chapter 3</a>
<ul>
<li class="chapter" data-level="A.1.1" data-path="appendexA.html"><a href="appendexA.html#exercise-3.1"><i class="fa fa-check"></i><b>A.1.1</b> Exercise 3.1:</a></li>
<li class="chapter" data-level="A.1.2" data-path="appendexA.html"><a href="appendexA.html#exercise-3.2"><i class="fa fa-check"></i><b>A.1.2</b> Exercise 3.2</a></li>
<li class="chapter" data-level="A.1.3" data-path="appendexA.html"><a href="appendexA.html#exercise-3.3"><i class="fa fa-check"></i><b>A.1.3</b> Exercise 3.3</a></li>
<li class="chapter" data-level="A.1.4" data-path="appendexA.html"><a href="appendexA.html#exercise-3.4"><i class="fa fa-check"></i><b>A.1.4</b> Exercise 3.4</a></li>
</ul></li>
<li class="chapter" data-level="A.2" data-path="appendexA.html"><a href="appendexA.html#chapter-8"><i class="fa fa-check"></i><b>A.2</b> Chapter 8</a>
<ul>
<li class="chapter" data-level="A.2.1" data-path="appendexA.html"><a href="appendexA.html#exercise-8.1"><i class="fa fa-check"></i><b>A.2.1</b> Exercise 8.1</a></li>
<li class="chapter" data-level="A.2.2" data-path="appendexA.html"><a href="appendexA.html#exercise-8.2"><i class="fa fa-check"></i><b>A.2.2</b> Exercise 8.2</a></li>
<li class="chapter" data-level="A.2.3" data-path="appendexA.html"><a href="appendexA.html#exercise-8.3"><i class="fa fa-check"></i><b>A.2.3</b> Exercise 8.3</a></li>
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<li class="chapter" data-level="A.3" data-path="appendexA.html"><a href="appendexA.html#chapter-14"><i class="fa fa-check"></i><b>A.3</b> Chapter 14</a>
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<li class="chapter" data-level="A.3.1" data-path="appendexA.html"><a href="appendexA.html#exercise-14.1"><i class="fa fa-check"></i><b>A.3.1</b> Exercise 14.1</a></li>
<li class="chapter" data-level="A.3.2" data-path="appendexA.html"><a href="appendexA.html#exercise-14.2"><i class="fa fa-check"></i><b>A.3.2</b> Exercise 14.2</a></li>
<li class="chapter" data-level="A.3.3" data-path="appendexA.html"><a href="appendexA.html#exercise-14.3"><i class="fa fa-check"></i><b>A.3.3</b> Exercise 14.3</a></li>
<li class="chapter" data-level="A.3.4" data-path="appendexA.html"><a href="appendexA.html#exercise-14.4"><i class="fa fa-check"></i><b>A.3.4</b> Exercise 14.4</a></li>
<li class="chapter" data-level="A.3.5" data-path="appendexA.html"><a href="appendexA.html#exercise-14.5"><i class="fa fa-check"></i><b>A.3.5</b> Exercise 14.5</a></li>
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<li class="chapter" data-level="A.4" data-path="appendexA.html"><a href="appendexA.html#chapter-17"><i class="fa fa-check"></i><b>A.4</b> Chapter 17</a>
<ul>
<li class="chapter" data-level="A.4.1" data-path="appendexA.html"><a href="appendexA.html#exercise-17.1"><i class="fa fa-check"></i><b>A.4.1</b> Exercise 17.1</a></li>
<li class="chapter" data-level="A.4.2" data-path="appendexA.html"><a href="appendexA.html#exercise-17.2"><i class="fa fa-check"></i><b>A.4.2</b> Exercise 17.2</a></li>
<li class="chapter" data-level="A.4.3" data-path="appendexA.html"><a href="appendexA.html#exercise-17.3"><i class="fa fa-check"></i><b>A.4.3</b> Exercise 17.3</a></li>
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<li class="chapter" data-level="A.5" data-path="appendexA.html"><a href="appendexA.html#chapter-20"><i class="fa fa-check"></i><b>A.5</b> Chapter 20</a>
<ul>
<li class="chapter" data-level="A.5.1" data-path="appendexA.html"><a href="appendexA.html#exercise-20.1"><i class="fa fa-check"></i><b>A.5.1</b> Exercise 20.1</a></li>
<li class="chapter" data-level="A.5.2" data-path="appendexA.html"><a href="appendexA.html#exercise-20.2"><i class="fa fa-check"></i><b>A.5.2</b> Exercise 20.2</a></li>
<li class="chapter" data-level="A.5.3" data-path="appendexA.html"><a href="appendexA.html#exercise-20.3"><i class="fa fa-check"></i><b>A.5.3</b> Exercise 20.3</a></li>
<li class="chapter" data-level="A.5.4" data-path="appendexA.html"><a href="appendexA.html#exercise-20.4"><i class="fa fa-check"></i><b>A.5.4</b> Exercise 20.4</a></li>
<li class="chapter" data-level="A.5.5" data-path="appendexA.html"><a href="appendexA.html#exercise-20.5"><i class="fa fa-check"></i><b>A.5.5</b> Exercise 20.5</a></li>
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<li class="chapter" data-level="A.6" data-path="appendexA.html"><a href="appendexA.html#chapter-23"><i class="fa fa-check"></i><b>A.6</b> Chapter 23</a>
<ul>
<li class="chapter" data-level="A.6.1" data-path="appendexA.html"><a href="appendexA.html#exercise-23.1"><i class="fa fa-check"></i><b>A.6.1</b> Exercise 23.1</a></li>
<li class="chapter" data-level="A.6.2" data-path="appendexA.html"><a href="appendexA.html#exercise-23.2"><i class="fa fa-check"></i><b>A.6.2</b> Exercise 23.2</a></li>
<li class="chapter" data-level="A.6.3" data-path="appendexA.html"><a href="appendexA.html#exercise-23.3"><i class="fa fa-check"></i><b>A.6.3</b> Exercise 23.3</a></li>
<li class="chapter" data-level="A.6.4" data-path="appendexA.html"><a href="appendexA.html#exercise-23.4"><i class="fa fa-check"></i><b>A.6.4</b> Exercise 23.4</a></li>
<li class="chapter" data-level="A.6.5" data-path="appendexA.html"><a href="appendexA.html#exercise-23.5"><i class="fa fa-check"></i><b>A.6.5</b> Exercise 23.5</a></li>
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<li class="chapter" data-level="A.7" data-path="appendexA.html"><a href="appendexA.html#chapter-28"><i class="fa fa-check"></i><b>A.7</b> Chapter 28</a>
<ul>
<li class="chapter" data-level="A.7.1" data-path="appendexA.html"><a href="appendexA.html#exercise-28.1"><i class="fa fa-check"></i><b>A.7.1</b> Exercise 28.1</a></li>
<li class="chapter" data-level="A.7.2" data-path="appendexA.html"><a href="appendexA.html#exercise-28.2"><i class="fa fa-check"></i><b>A.7.2</b> Exercise 28.2</a></li>
<li class="chapter" data-level="A.7.3" data-path="appendexA.html"><a href="appendexA.html#exercise-28.3"><i class="fa fa-check"></i><b>A.7.3</b> Exercise 28.3</a></li>
<li class="chapter" data-level="A.7.4" data-path="appendexA.html"><a href="appendexA.html#exercise-28.4"><i class="fa fa-check"></i><b>A.7.4</b> Exercise 28.4</a></li>
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<li class="chapter" data-level="A.8" data-path="appendexA.html"><a href="appendexA.html#chapter-31"><i class="fa fa-check"></i><b>A.8</b> Chapter 31</a>
<ul>
<li class="chapter" data-level="A.8.1" data-path="appendexA.html"><a href="appendexA.html#exercise-31.1"><i class="fa fa-check"></i><b>A.8.1</b> Exercise 31.1</a></li>
<li class="chapter" data-level="A.8.2" data-path="appendexA.html"><a href="appendexA.html#exercise-31.2"><i class="fa fa-check"></i><b>A.8.2</b> Exercise 31.2</a></li>
<li class="chapter" data-level="A.8.3" data-path="appendexA.html"><a href="appendexA.html#exercise-31.3"><i class="fa fa-check"></i><b>A.8.3</b> Exercise 31.3</a></li>
<li class="chapter" data-level="A.8.4" data-path="appendexA.html"><a href="appendexA.html#exercise-31.4"><i class="fa fa-check"></i><b>A.8.4</b> Exercise 31.4</a></li>
<li class="chapter" data-level="A.8.5" data-path="appendexA.html"><a href="appendexA.html#exercise-31.5"><i class="fa fa-check"></i><b>A.8.5</b> Exercise 31.5</a></li>
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<li class="chapter" data-level="A.9" data-path="appendexA.html"><a href="appendexA.html#chapter-34"><i class="fa fa-check"></i><b>A.9</b> Chapter 34</a>
<ul>
<li class="chapter" data-level="A.9.1" data-path="appendexA.html"><a href="appendexA.html#exercise-34.1"><i class="fa fa-check"></i><b>A.9.1</b> Exercise 34.1</a></li>
<li class="chapter" data-level="A.9.2" data-path="appendexA.html"><a href="appendexA.html#exercise-34.2"><i class="fa fa-check"></i><b>A.9.2</b> Exercise 34.2</a></li>
<li class="chapter" data-level="A.9.3" data-path="appendexA.html"><a href="appendexA.html#exercise-34.3"><i class="fa fa-check"></i><b>A.9.3</b> Exercise 34.3</a></li>
<li class="chapter" data-level="A.9.4" data-path="appendexA.html"><a href="appendexA.html#exercise-34.4"><i class="fa fa-check"></i><b>A.9.4</b> Exercise 34.4</a></li>
<li class="chapter" data-level="A.9.5" data-path="appendexA.html"><a href="appendexA.html#exercise-33.5"><i class="fa fa-check"></i><b>A.9.5</b> Exercise 33.5</a></li>
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<li class="chapter" data-level="B" data-path="uncertainty_derivation.html"><a href="uncertainty_derivation.html"><i class="fa fa-check"></i><b>B</b> Uncertainty derivation</a>
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<li class="chapter" data-level="B.1" data-path="uncertainty_derivation.html"><a href="uncertainty_derivation.html#propagation-of-error-for-addition-and-subtraction"><i class="fa fa-check"></i><b>B.1</b> Propagation of error for addition and subtraction</a></li>
<li class="chapter" data-level="B.2" data-path="uncertainty_derivation.html"><a href="uncertainty_derivation.html#propagation-of-error-for-multiplication-and-division"><i class="fa fa-check"></i><b>B.2</b> Propagation of error for multiplication and division</a></li>
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<h1><span class="header-section-number">Chapter 1</span> Background mathematics<a href="Chapter_1.html#Chapter_1" class="anchor-section" aria-label="Anchor link to header"></a></h1>
<p>There are at least two types of mathematical challenges that come with first learning statistics.
The first challenge is simply knowing the background mathematics upon which many statistical tools rely.
Fortunately, while the <em>theory</em> underlying statistical techniques does rely on some quite advanced mathematics <span class="citation">(e.g., see <a href="#ref-Mclean1991" role="doc-biblioref">Mclean et al., 1991</a>; <a href="#ref-Miller2004" role="doc-biblioref">Miller & Miller, 2004</a>; <a href="#ref-Rencher2000" role="doc-biblioref">Rencher, 2000</a>)</span>, the <em>application</em> of standard statistical tools usually does not.
This book focuses on the application of statistical techniques, so all that is required is a background in some fundamental mathematical concepts such as mathematical operations (addition, subtraction, multiplication, division, and exponents), simple algebra, and probability.
This chapter will review mathematical operations and the symbols used to communicate them.</p>
<p>The second mathematical challenge that students face when learning statistics for the first time is a bit more subtle.
Students with no statistical background sometimes have an expectation that statistics will be similar to previously learnt mathematical topics such as algebra, geometry, or trigonometry.
In some ways, this is the case, but in a lot of ways statistics is a much different way of thinking than any of these topics.
A lot of mathematical subjects focus on questions that have very clear right or wrong answers (or, at least, this is how they are often taught).
If, for example, we are given the lengths of two sides of a right triangle, then we might be asked to calculate the hypotenuse of the triangle using Pythagorean theorem (<span class="math inline">\(a^{2} + b^{2} = c^{2}\)</span>, where c is the hypotenuse).
If we know the length of the two sides, then the length of the hypotenuse has a clear correct answer (at least, on a Euclidean plane).
In statistics, answers are not always so clear-cut.
Statistics, by its very nature, deals with uncertainty.
While all of the standard rules of mathematics still apply, statistical questions such as, ‘Can I use this statistical test on my data?’, ‘Do I have a large enough sample size?’, or even ‘Is my hypothesis well-supported?’ often do not have unequivocal ‘correct’ answers.
Being a good statistician often means making well-informed, but ultimately at least somewhat subjective, judgements about how to make inferences from data.</p>
<p>For now, we will move on to looking at numbers and operations, logarithms, and order of operations.
These topics will be relevant throughout the book, so it is important to understand them and be able to apply them when doing calculations.</p>
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<h2><span class="header-section-number">1.1</span> Numbers and operations<a href="Chapter_1.html#numbers-and-operations" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>Calculating statistics and reading statistical output requires some knowledge of numbers and basic mathematical operations.
This section is a summary of the basic mathematical tools that will be used in introductory statistics.
Much of this section is inspired by <span class="citation">Courant et al. (<a href="#ref-Courant1996" role="doc-biblioref">1996</a>)</span> and chapter 2 of <span class="citation">Pastor (<a href="#ref-Pastor2008" role="doc-biblioref">2008</a>)</span>.
This section will focus on only the numbers and mathematical operations relevant to this book.
The objective here is to present some very well-known ideas in an interesting way, and to intermix them with bits of information that might be new and interesting.
For doing statistics, what you really <em>need</em> to know here are the operations and the notation, that is, how operations such as addition, multiplication, and exponents are calculated and represented mathematically.</p>
<p>We can start with the <em>natural</em> numbers, which are the kinds of numbers that can be counted using fingers, toothpicks, pebbles, or any discrete sets of objects.</p>
<p><span class="math display">\[1, 2, 3, 4, 5, 6, 7, 8, ...\]</span></p>
<p>There are an infinite number of natural numbers (we can represent the set of all of them using the symbol <span class="math inline">\(\mathbb{N}\)</span>).
For any given natural number, we can always find a higher natural number using the operation of addition.
For example, a number higher than 5 can be obtained by simply adding 1 to it,</p>
<p><span class="math display">\[5 + 1 = 6.\]</span></p>
<p>This is probably not that much of a revelation, but it highlights why the natural numbers are countably infinite (for any number you can think of, <span class="math inline">\(N\)</span>, there is always a higher number <span class="math inline">\(N + 1\)</span>).
It also leads to a reminder about two other important mathematical symbols for this book (in addition to <span class="math inline">\(+\)</span>, which indicates addition), greater than (<span class="math inline">\(>\)</span>) and less than (<span class="math inline">\(<\)</span>).
We know that the number 6 is greater than 5, and we express this mathematically as the <strong>inequality</strong>, <span class="math inline">\(6 > 5\)</span>.
Note that the large end of the inequality faces the higher number, while the pointy end (i.e., the smaller end) faces the lower number.
Inequalities are used regularly in statistics, e.g., to indicate when a probability of something is less than a given value (e.g., <span class="math inline">\(P < 0.05\)</span>, which can be read ‘P is less than 0.05’).
We might also use the symbols <span class="math inline">\(\geq\)</span> or <span class="math inline">\(\leq\)</span> to indicate when something is greater than or equal to (<span class="math inline">\(\geq\)</span>) or less than or equal to (<span class="math inline">\(\leq\)</span>) a particular value.
For example, <span class="math inline">\(x \geq 10\)</span> indicates that some number <span class="math inline">\(x\)</span> has a value of 10 or higher.</p>
<p>Whenever we add one natural number to another natural number, the result is another natural number, a sum (e.g., <span class="math inline">\(5 + 1 = 6\)</span>).
If we want to go back from the sum to one of the values being summed (i.e., get from <span class="math inline">\(6\)</span> to <span class="math inline">\(5\)</span>), then we need to subtract,</p>
<p><span class="math display">\[6 - 1 = 5.\]</span></p>
<p>This operation is elementary mathematics, but a subtle point that is often missed is that the introduction of subtraction creates the need for a broader set of numbers than the natural numbers.
We call this broader set of numbers the <em>integers</em> (we can represent these using the symbol <span class="math inline">\(\mathbb{Z}\)</span>).
If, for example, we want to subtract 5 from 1, we get a number that cannot be represented on our fingers,</p>
<p><span class="math display">\[1 - 5 = -4.\]</span></p>
<p>The value <span class="math inline">\(-4\)</span> is an integer (but <em>not</em> a natural number).
Integers include 0 and all negative whole numbers,</p>
<p><span class="math display">\[..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ...\]</span></p>
<p>Whenever we add or subtract integers, the result is always another integer.</p>
<p>Now, suppose we wanted to add the same value up multiple times.
For example,</p>
<p><span class="math display">\[2 + 2 + 2 + 2 + 2 + 2 = 12.\]</span></p>
<p>The number 2 is being added 6 times in the equation above to get a value of 12.
But we can represent this sum more easily using the operation of multiplication,</p>
<p><span class="math display">\[2 \times 6 = 12.\]</span>
The 6 in the equation just represents the number of times that 2 is being added up.
The equation can also be written as <span class="math inline">\(2(6) = 12\)</span>, or sometimes, 2<code>*</code>6 = 12 (i.e., the asterisk is sometimes used to indicate multiplication).
Parentheses (i.e., round brackets) indicate multiplication when no other symbol separates them from a number.
This rule also applies to numbers that come immediately before variables.
For example, <span class="math inline">\(2x\)</span> can be interpreted as <em>two times x</em>.
When multiplying integers, we always get another integer.
Multiplying two positive numbers always equals another positive number (e.g., <span class="math inline">\(2 \times 6 = 12\)</span>).
Multiplying a positive and a negative number equals a negative number (e.g., <span class="math inline">\(-2 \times 6 = -12\)</span>).
And multiplying two negative numbers equals a positive number (e.g., <span class="math inline">\(-2 \times -6 = 12\)</span>).
There are multiple ways of thinking about why this last one is true <span class="citation">(see, e.g., <a href="#ref-Askey1999" role="doc-biblioref">Askey, 1999</a> for one explanation)</span>, but for now we can take it as a given.</p>
<p>As with addition and subtraction, we need an operation that can go back from multiplied values (the product) to the numbers being multiplied.
In other words, if we multiply to get <span class="math inline">\(2 \times 6 = 12\)</span> (where 12 is the product), then we need something that goes back from 12 to 2.
Division allows us to do this, such that <span class="math inline">\(12 \div 6 = 2\)</span>.
In statistics, the symbol <span class="math inline">\(\div\)</span> is rarely used, and we would more often express the calculation as either <span class="math inline">\(12/6 = 2\)</span> or,</p>
<p><span class="math display">\[\frac{12}{6} = 2.\]</span></p>
<p>As with subtraction, there is a subtle point that the introduction of division requires a new set of numbers.
If instead of dividing 6 into 12, we divided 12 into 6,</p>
<p><span class="math display">\[\frac{6}{12} = \frac{1}{2} = 0.5.\]</span></p>
<p>We now have a number that is not an integer.
We therefore need a new broader set of numbers, the <em>rational</em> numbers (we can represent these using the symbol <span class="math inline">\(\mathbb{Q}\)</span>).
The rationals include all numbers that can be expressed as a <em>ratio</em> of integers.
That is, <span class="math inline">\(p / q\)</span>, where both <span class="math inline">\(p\)</span> and <span class="math inline">\(q\)</span> are in the set <span class="math inline">\(\mathbb{Z}\)</span>.</p>
<p>We have one more set of operations relevant for introductory statistics.
Recall that we introduced <span class="math inline">\(2 \times 6\)</span> as a way to represent <span class="math inline">\(2 + 2 + 2 + 2 + 2 + 2\)</span>.
We can apply the same logic to multiplying a number multiple times.
For example, we might want to multiply the number 2 by itself 4 times,</p>
<p><span class="math display">\[2 \times 2 \times 2 \times 2 = 16.\]</span></p>
<p>We can represent this more compactly using an <strong>exponent</strong>, which is written as a superscript,</p>
<p><span class="math display">\[2^{4} = 16.\]</span></p>
<p>The 4 in the equation above indicates that the 2 should be multiplied 4 times to get 16.
Sometimes this is also represented by a caret in writing or code, such that 2^4 = 16.
Very occasionally, some authors will use two asterisks in a row, 2**4 = 16, probably because this is how exponents are represented in some statistical software and programming languages.
One quick note that can be confusing at first is that a negative in the exponent indicates a reciprocal.
For example,</p>
<p><span class="math display">\[2^{-4} = \frac{1}{16}.\]</span></p>
<p>This can sometimes be useful for representing the reciprocal of a number or unit in a more compact way than using a fraction (we will come back to this in <a href="Chapter_6.html#Chapter_6">Chapter 6</a>).</p>
<p>As with addition and subtraction, and multiplication and division, we also need an operation to get back from the exponentiated value to the original number.
That is, for <span class="math inline">\(2^{4} = 16\)</span>, there should be an operation that gets us back from 16 to 2.
We can do this using the <strong>root</strong> of an equation,</p>
<p><span class="math display">\[\sqrt[4]{16} = 2.\]</span></p>
<p>The number under the radical symbol <span class="math inline">\(\sqrt{}\)</span> (in this case 16) is the one that we are taking the root of, and the index (in this case 4) is the root that we are calculating.
When the index is absent, we assume that it is 2 (i.e., a square root),</p>
<p><span class="math display">\[\sqrt[2]{16} = \sqrt{16} = 4.\]</span></p>
<p>Note that <span class="math inline">\(4^{2} = 16\)</span> (i.e., 4 squared equals 16).</p>
<p>Instead of using the radical symbol, we could also use a fraction in the exponent.
That is, instead of writing <span class="math inline">\(\sqrt[4]{16} = 2\)</span>, we could write <span class="math inline">\(16^{1/4} = 2\)</span> or <span class="math inline">\(16^{1/2} = 4\)</span>.
In statistics, however, the <span class="math inline">\(\sqrt{}\)</span> is more often used.
Either way, this yet again creates the need for an even broader set of numbers.
This is because expressions such as <span class="math inline">\(\sqrt{2}\)</span> do not equal any rational number.
In other words, there are no integers <span class="math inline">\(p\)</span> and <span class="math inline">\(q\)</span> such that their <em>ratio</em>, <span class="math inline">\(p/q = \sqrt{2}\)</span> (the proof for why is very elegant!).
Consequently, we can say that <span class="math inline">\(\sqrt{2}\)</span> is <em>irrational</em> (not in the colloquial sense of being illogical or unreasonable, but in the technical sense that it cannot be represented as a ratio of two integers).
Irrational numbers cannot be represented as a ratio of integers, or with a finite or repeating decimal.
Remarkably, the set of irrational numbers is larger than the set of rational numbers (i.e., rational numbers are countably infinite, while irrational numbers are uncountably infinite, and there are more irrationals; you do not need to know this or even believe it, but it is true!).</p>
<p>Perhaps the most famous irrational number is <span class="math inline">\(\pi\)</span>, which appears throughout science and mathematics and is most commonly introduced as the ratio of a circle’s circumference to its diameter.
Its value is <span class="math inline">\(\pi \approx 3.14159\)</span>, where the symbol <span class="math inline">\(\approx\)</span> means ‘approximately.’
Actually, the decimal expansion of <span class="math inline">\(\pi\)</span> is infinite and non-repeating; the decimals go on forever and never repeat themselves in a predictable pattern.
As of 2019, over 31 trillion (i.e., 31000000000000) decimals of <span class="math inline">\(\pi\)</span> have been calculated <span class="citation">(<a href="#ref-Yee2019" role="doc-biblioref">Yee, 2019</a>)</span>.</p>
<p>The rational and irrational numbers together comprise a set of numbers called <em>real</em> numbers (we can represent these with the symbol <span class="math inline">\(\mathbb{R}\)</span>), and this is where we will stop.
This story of numbers and operations continues with imaginary and complex numbers <span class="citation">(<a href="#ref-Courant1996" role="doc-biblioref">Courant et al., 1996</a>; <a href="#ref-Pastor2008" role="doc-biblioref">Pastor, 2008</a>)</span>, but these are not necessary for introductory statistics.</p>
</div>
<div id="logarithms" class="section level2 hasAnchor" number="1.2">
<h2><span class="header-section-number">1.2</span> Logarithms<a href="Chapter_1.html#logarithms" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>There is one more important mathematical operation to mention that is relevant to introductory statistics.
Logarithms are important functions, which will appear in multiple places (e.g., statistical transformations of variables).
A logarithm tells us the exponent to which a number needs to be raised to get another number.
For example,</p>
<p><span class="math display">\[10^{3} = 1000.\]</span></p>
<p>Verbally, 10 raised to the power of 3 equals 1000.
In other words, we need to raise 10 to the power of 3 to get a value of 1000.
We can express this using a logarithm,</p>
<p><span class="math display">\[\log_{10}\left(1000\right) = 3.\]</span></p>
<p>Again, the same relationship is expressed in <span class="math inline">\(10^{3} = 1000\)</span> and <span class="math inline">\(\log_{10}(1000) = 3\)</span>.
For the latter, we might say that the base 10 logarithm of 1000 is 3.
This is actually extremely useful in mathematics and statistics.
Mathematically, logarithms have the very useful property,</p>
<p><span class="math display">\[\log_{10}(ab) = \log_{10}(a) + \log_{10}(b).\]</span></p>
<p>Historically, this has been used to make calculations easier by converting multiplication to addition <span class="citation">(<a href="#ref-Stewart2008" role="doc-biblioref">Stewart, 2008</a>)</span>.
In statistics, and across the biological and environmental sciences, we often use logarithms when we want to represent something that changes exponentially on a more convenient scale.
For example, suppose that we wanted to illustrate the change in global CO<span class="math inline">\(_{2}\)</span> emissions over time <span class="citation">(<a href="#ref-Friedlingstein2022" role="doc-biblioref">Friedlingstein et al., 2022</a>)</span>.
We could show year on the x-axis and emissions in billions of tonnes of CO<span class="math inline">\(_{2}\)</span> on the y-axis (Figure 1.1).</p>
<div class="figure"><span style="display:block;" id="fig:unnamed-chunk-2"></span>
<img src="bookdown-demo_files/figure-html/unnamed-chunk-2-1.png" alt="A plot showing an exponential increase in global carbon dioxide emissions over time" width="100%" />
<p class="caption">
Figure 1.1: Global carbon dioxide emissions 1750–2021.
</p>
</div>
<p>We can see from Figure 1.1 that global CO<span class="math inline">\(_{2}\)</span> emissions go up exponentially over time, but this exponential relationship means that the y-axis has to cover a large range of values.
This makes it difficult to see what is actually happening in the first 100 years.
Are CO<span class="math inline">\(_{2}\)</span> emissions increasing from 1750 to 1850, or do they stay about the same?
If instead of plotting billions of tonnes of CO<span class="math inline">\(_{2}\)</span> on the y-axis, we plotted the logarithm of these values, then the pattern in the first 100 years becomes a bit clearer (Figure 1.2).</p>
<div class="figure"><span style="display:block;" id="fig:unnamed-chunk-3"></span>
<img src="bookdown-demo_files/figure-html/unnamed-chunk-3-1.png" alt="A plot showing linear increase in global carbon dioxide emissions over time, with emissions plotted on a natural log scale." width="100%" />
<p class="caption">
Figure 1.2: Natural logarithm of global carbon dioxide emissions 1750–2021.
</p>
</div>
<p>It appears from the logged data in Figure 1.2 that global CO<span class="math inline">\(_{2}\)</span> emissions were indeed increasing from 1750 to 1850.
Note that Figure 1.2 presents the <em>natural logarithm</em> of CO<span class="math inline">\(_{2}\)</span> emissions on the y-axis.
The natural logarithm uses Euler’s number, <span class="math inline">\(e \approx 2.718282\)</span>, as a base.
Euler’s number <span class="math inline">\(e\)</span> is an irrational number (like <span class="math inline">\(\pi\)</span>), which corresponds to the intrinsic rate of increase of a population’s size in ecology <span class="citation">(<a href="#ref-Gotelli2001" role="doc-biblioref">Gotelli, 2001</a>)</span>, or, in banking, interest compounded continuously (like <span class="math inline">\(\pi\)</span>, <span class="math inline">\(e\)</span> actually shows up in a lot of different places throughout science and mathematics).
We probably could have just as easily used 10 as a base, but <span class="math inline">\(e\)</span> is usually the default base to use in science (bases 10 or 2 are also often used).
Note that we can convert back to the non-logged scale by raising numbers to the power of <span class="math inline">\(e\)</span>.
For example, <span class="math inline">\(e^{-4} \approx 0.018\)</span>, <span class="math inline">\(e^{-2} \approx 0.135\)</span>, <span class="math inline">\(e^{0} = 1\)</span>, and <span class="math inline">\(e^{2} = 7.390\)</span>.</p>
</div>
<div id="order-of-operations" class="section level2 hasAnchor" number="1.3">
<h2><span class="header-section-number">1.3</span> Order of operations<a href="Chapter_1.html#order-of-operations" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>Every once in a while, a maths problem like the one below seems to go viral online,</p>
<p><span class="math display">\[x = 8 \div 2\left(2+2\right).\]</span></p>
<p>Depending on the order in which calculations are made, some people will conclude that <span class="math inline">\(x = 16\)</span>, while others conclude that <span class="math inline">\(x = 1\)</span> <span class="citation">(<a href="#ref-Chernoff2022" role="doc-biblioref">Chernoff & Zazkis, 2022</a>)</span>.
The confusion is not caused by the above calculation being difficult, but by peoples’ differences in interpreting the rules for the order by which calculations should be carried out.
If we first divide 8/2 to get 4, then multiply by (2 + 2), we get 16.
If we first multiply 2 by (2 + 2) to get 8, then divide, we get 1.
The truth is that even if there is a ‘right’ answer here <span class="citation">(<a href="#ref-Chernoff2022" role="doc-biblioref">Chernoff & Zazkis, 2022</a>)</span>, the equation could be written more clearly.
We might, for example, rewrite the above to more clearly express the intended order of operations,</p>
<p><span class="math display">\[x = \frac{8}{2}\left(2 + 2\right) = 16.\]</span></p>
<p>We could write it a different way to express a different intended order of operations,</p>
<p><span class="math display">\[x = \frac{8}{2(2+2)} = 1.\]</span></p>
<p>The key point is that the order in which operations are calculated matters, so it is important to write equations clearly, and to know the order of operations to calculate an answer correctly.
By convention, there are some rules for the order in which calculations should proceed.</p>
<ol style="list-style-type: decimal">
<li>Anything within parentheses should always be calculated first.</li>
<li>Exponents and radicals should be applied second.</li>
<li>Multiplication and division should be applied third.</li>
<li>Addition and subtraction should be done last.</li>
</ol>
<p>These conventions are not really rooted in anything fundamental about numbers or operations (i.e., we made these rules up), but there is a logic to them.
First, parentheses are a useful tool for being unequivocal about the order of operations.
We could, for example, always be completely clear about the order to calculate by writing something like <span class="math inline">\((8/2) \times (2+2)\)</span> or <span class="math inline">\(8 / (2(2 + 2))\)</span>, although this can get a bit messy.
Second, rules 2–4 are ordered by the magnitude of operation effects; for example, exponents have a bigger effect than multiplication, which has a bigger effect than addition.
In general, however, these are just standard conventions that need to be known for reading and writing mathematical expressions.
In this book, you will not see something ambiguous like <span class="math inline">\(x = 8 \div 2\left(2+2\right)\)</span>, but you should be able to correctly calculate something like this,</p>
<p><span class="math display">\[x = 3^{2} + 2\left(1 + 3\right)^{2} - 6 \times 0.\]</span></p>
<p>First, remember that parentheses come first, so we can rewrite the above,</p>
<p><span class="math display">\[x = 3^{2} + 2\left(4\right)^{2} - 6 \times 0.\]</span></p>
<p>Exponents come next, so we can calculate those,</p>
<p><span class="math display">\[x = 9 + 2\left(16\right) - 6 \times 0.\]</span></p>
<p>Next comes multiplication and division,</p>
<p><span class="math display">\[x = 9 + 32 - 0.\]</span></p>
<p>Lastly, we calculate addition and subtraction,</p>
<p><span class="math display">\[x = 41.\]</span></p>
<p>In this book, you will very rarely need to calculate something with this many different steps.
But you will often need to calculate equations like the one below,</p>
<p><span class="math display">\[x = 20 + 1.96 \times 2.1.\]</span></p>
<p>It is important to remember to multiply <span class="math inline">\(1.96 \times 2.1\)</span> <em>before</em> adding 20.
Getting the order of operations wrong will usually result in the calculation being completely off.</p>
<p>One last note is that when operations are above or below a fraction, or below a radical, then parentheses are implied.
For example, we might have something like the fraction below,</p>
<p><span class="math display">\[x = \frac{2^{2} + 1}{3^{2} +2}.\]</span></p>
<p>Although rules 2–4 still apply, it is implied that there are parentheses around both the top (numerator) and bottom (denominator), so you can always read the above equation like this,</p>
<p><span class="math display">\[x = \frac{\left(2^{2} + 1\right)}{\left(3^{2} + 2\right)} = \frac{\left(4 + 1\right)}{\left(9 + 2\right)} = \frac{5}{11}.\]</span></p>
<p>Similarly, anything under the <span class="math inline">\(\sqrt{}\)</span> can be interpreted as being within parentheses.
For example,</p>
<p><span class="math display">\[x = \sqrt{3 + 4^{2}} = \sqrt{\left(3 + 4^{2} \right)} \approx 4.47.\]</span></p>
<p>This can take some getting used to, but with practice, it will become second nature to read equations with the correct order of operations.</p>
</div>
</div>
<h3>References<a href="references.html#references" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<div id="refs" class="references csl-bib-body hanging-indent" line-spacing="2">
<div id="ref-Askey1999" class="csl-entry">
Askey, R. (1999). <span class="nocase">Why does a negative x a negative = a positive?</span> <em>American Educator</em>, <em>23</em>(3), 4–5.
</div>
<div id="ref-Chernoff2022" class="csl-entry">
Chernoff, E. J., & Zazkis, R. (2022). <span class="nocase">The simple reason a viral math equation stumped the internet</span>. <em>The Conversation</em>. <a href="https://theconversation.com/the-simple-reason-a-viral-math-equation-stumped-the-internet-176518">https://theconversation.com/the-simple-reason-a-viral-math-equation-stumped-the-internet-176518</a>
</div>
<div id="ref-Courant1996" class="csl-entry">
Courant, R., Robbins, H., & Stewart, I. (1996). <em><span class="nocase">What is Mathematics?</span></em> (2nd ed., p. 566). Oxford University Press, Oxford, UK.
</div>
<div id="ref-Friedlingstein2022" class="csl-entry">
Friedlingstein, P., O’Sullivan, M., Jones, M. W., Andrew, R. M., Gregor, L., Hauck, J., Le Quéré, C., Luijkx, I. T., Olsen, A., Peters, G. P., Peters, W., Pongratz, J., Schwingshackl, C., Sitch, S., Canadell, J. G., Ciais, P., Jackson, R. B., Alin, S. R., Alkama, R., … Zheng, B. (2022). Global carbon budget 2022. <em>Earth System Science Data</em>, <em>14</em>(11), 4811–4900. <a href="https://doi.org/10.5194/essd-14-4811-2022">https://doi.org/10.5194/essd-14-4811-2022</a>
</div>
<div id="ref-Gotelli2001" class="csl-entry">
Gotelli, N. J. (2001). <em><span class="nocase">A Primer of Ecology</span></em> (3rd ed., p. 265). Sinauer Associates, Inc., Sunderland, Massachusetts, USA.
</div>
<div id="ref-Mclean1991" class="csl-entry">
Mclean, R. A., Sanders, W. L., & Stroup, W. W. (1991). <span class="nocase">A unified approach to mixed linear models</span>. <em>American Statistician</em>, <em>45</em>(1), 54–64.
</div>
<div id="ref-Miller2004" class="csl-entry">
Miller, I., & Miller, M. (2004). <em><span class="nocase">John E. Freund’s mathematical statistics</span></em> (7th ed., p. 614). Pearson Prentice Hall, Upper Saddle River, New Jersey, USA.
</div>
<div id="ref-Pastor2008" class="csl-entry">
Pastor, J. (2008). <em><span class="nocase">Mathematical Ecology of Populations and Ecosystems</span></em> (pp. 1–50). John Wiley & Sons, Inc., West Sussex, England.
</div>
<div id="ref-Rencher2000" class="csl-entry">
Rencher, A. C. (2000). <em><span class="nocase">Linear Models in Statistics</span></em> (p. 578). John Wiley & Sons, Inc., Provo, Utah, USA.
</div>
<div id="ref-Stewart2008" class="csl-entry">
Stewart, I. (2008). <em><span class="nocase">Taming the Infinite</span></em> (p. 384). Quercus, London, UK.
</div>
<div id="ref-Yee2019" class="csl-entry">
Yee, A. J. (2019). <em><span class="nocase">Google Cloud Topples the Pi Record</span></em>. <a href="http://www.numberworld.org/blogs/2019_3_14_pi_record/">http://www.numberworld.org/blogs/2019_3_14_pi_record/</a>
</div>
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