2  Basic Notation

2.1 Lists

Most of the time, you will see \(x\) denote a list of values (i.e. a variable in a data table).

For example, if \(x\) was the list of numerical values \(a\) to \(g\), we can write it as:

\(x = [a, b, c, d, e, f, g]\)

Then, \(x_i\) means the \(i^{th}\) value in the list of \(x\), so

\(x_1 = a\) and \(x_2 = b\) and \(x_7 = g\).

2.2 \(n\)

In regards to a data table or list of values, \(n\) stands for the number of rows or data points that are in the data table or list (we will learn this as the sample size later on)

So, for list \(x\), \(n = 7\).

Adding on, \(x_n\) would mean the last value in the list \(x\) (since there are only \(n\) values in \(x\))

2.3 Summation (\(\Sigma\))

You will also see the greek letter \(\Sigma\) in formulas. Usage of this sign means that we are using summation notation.

If we want the sum of all numbers from 1 to 7, we would write it as,

\[\sum_{i=1}^7 i\]

We interpret this as,

  1. Start from \(i = 1\), evaluate the expression, which is \(i\).

  2. Keep our evaluated expression to the side and get ready to add the other values to it, so

\[1 + \cdots\]

  1. Now go the next numbers until we get to \(7\) (the number on the top of the \(\Sigma\)) So moving onto \(i = 2\), we end up with \[1 + 2 + \cdots\] And with \(i = 3\), we end up with \[1 + 2 + 3 + \cdots\]

  2. When we get to the end of it (when we reach \(i = 7\)), we have the expanded form of the summation. \[1 + 2 + 3+ 4 + 5 + 6 + 7\]

2.4 Other notation

  • \(\bar x\): The “line” on top of \(x\) means the mean of \(x\). If we had \(\bar a\), I would be asking for the mean of \(a\).
    • Pronounced “x bar”
  • \(\hat p\): The “hat” on top of \(p\) means the estimate of \(p\). If we had \(\hat x\), I would be asking for the estimate of \(x\).
    • Pronounced “p hat”

2.5 Other commonly used symbols

  • \(N\): population size
  • \(p\): proportion, probability, or p-value
  • \(\mu\): population mean (true mean)
  • \(\bar x\): sample mean
  • \(\sigma\): population standard deviation (true standard deviation)
  • \(s_x\): sample standard deviation (of x), so \(s_y\) is the sample standard deviation of \(y\)
  • \(SE_{\bar x}\): Standard error of the sample mean (of x), the estimate of the standard deviation of the sample mean.