More math formatting experiments.

.gitignore

@@ -1,2 +1,3 @@
 _build/
-.DS_Store
+.DS_Store
+.vscode

how_it_works/10-StatsRefresher.md

@@ -19,17 +19,26 @@ exports:
     article_type: Report
 numbering:
   code: false
-  headings: true
+  headings: false
 ---
 
-The arithmetic mean:
-$$ \bar{x} = \sum_{i=1}^n x_i $$
+# The arithmetic mean
+The arithmetic mean $\bar{x}$ for $n$ measurements indexed $i = 1\ldots n$ is
+$$ \bar{x} = \dfrac{1}{n}\sum_{i=1}^n x_i \label{eq:ArithmeticMean}$$
 
-The variance
-$$ \sigma^2 = \sum_{i=1}^n \big( x_i - \bar{x}\big)^2 $$
 
-The standard deviation
+# The variance
+The variance $\sigma^2$ is a measure of how scattered about the mean the data points are.
+$$ \sigma^2 = \dfrac{1}{n-1}\sum_{i=1}^n \big( x_i - \bar{x}\big)^2 \label{eq:Variance} $$
+where $\bar{x}$ is the mean defined in equation [](#eq:ArithmeticMean). The variance is tricky, since it has the same units as your measurements, squared.
+
+
+# The standard deviation
+The standard deviation also measures how your data points scatter around the mean, but this time in units that you actually understand.
 $$ \sigma = \sqrt{\sigma^2} $$ 
+where $\sigma^2$ is the variance defined in equation [](#eq:Variance). 
+
+# The standard error
+The standard error tells you how well you know the mean.  It gets smaller as $n$ gets bigger, reflecting your improved knowledge of the mean as you make more measurements.
+$$ \sigma_{\bar{x}} = \dfrac{\sigma}{\sqrt{n}} $$
 
-The standard error
-$$ \sigma_{\bar{x}} = \dfrac{\sigma}{\sqrt{n}} $$