Standard Deviation Formula in Statistics

Question

I'm doing an introductory statistics course in Coursera and I saw a formula for standard deviation, which is apparently the difference between the 3rd and 1st quartile, or the spread (please correct me if I didn't define that correctly).

This is the formula:

I tried to google some things in response to this formula such as: "What is sideways E in math?" and "Standard deviation formula".

Let's just say that I am a complete math noob and I didn't get very far. My questions are:

1. What is the sideways E?
2. What is "n" above the E and the "i=1" below the E?
3. What does the "i" next to the "x" in the parenthesis mean?
4. What does it mean when there is a dash above the "x" inside the parenthesis?

Answer 1

Welcome. The difference between the first and third quartiles is the interquartile range. When we have cumulative frequency data (often box plots) IQR is used (sometimes) in favour of standard deviation. However, standard deviation is a decidedly different value.

In the context of population data, the standard deviation is defined as the square root of the variance. The variance (of a random variable or of a data population) is the mean of the squared difference between the mean of the data set divided by the total number of data points.

I will do an example and explain all notation.

Given finite discrete data, we can count the data points using index notation: $x_i$ for $1\le i\le n$. For example, $\{25.0,-129,36\}$ could be enumerated as $\{x_i\}_{i=1}^3$ where $x_1=25.0,x_2=-129,x_3=36$ ($n=3$ because of three data points). $\overline{x}$ is the common notation for the mean of the dataset, here that is $(1/3)(25.0-129+36)=-22.667$ (it is common to round of decimal places in numerical statistics). We then compute the variance as the sum of squared differences from the mean - a deviation, or difference, could be $(x_i-\overline{x})^2=(x_i+22.667)^2$ for any data point $x_i$ (e.g. $x_1=25$).

This sum becomes (that is all that $\Sigma$ means: a sum over the terms indexed by $i$): $$\sum_{i=1}^n(x_i-\overline{x})^2=(25+22.667)^2+(-129+22.667)^2+(36+22.667)^2=17,020.667$$The population variance, usually denoted $\sigma^2$, is: $$\sigma^2=\frac{1}{n}\sum_{i=1}^n(x_i-\overline{x})^2=\frac{1}{3}(17,020.667)=5673.556$$Notice that this is a really large number. That's because variance measures spread, and the spread in the data set I gave is huge (in order to exaggerate the point). The standard deviation is also a measure of spread (hmm, more correctly a measure of central tendency) and we take it as the square root of the variance: $$\sigma=\sqrt{\sigma^2}=\sqrt{\mathrm{Variance}}=\sqrt{5673.556}=75.323$$People often interpret the standard deviation as representative of the average value a data point might be from the mean (e.g., if a data point is further than two standard deviations away from the mean we often call it an outlier).

Why did I use "$\sigma$" and not "$s$"? It is common to use Greek letters (like $\sigma$) for population data and Roman letters (like $s$) for sample data. The distinction is made here because unfortunately there is a notion of sample mean, variance, standard deviation, which are often taken with a division by $n-1$ rather than a division by $n$ (this is called Bessel's correction, and is done because a sample isn't necessarily representative of the general population).

Answer 2

1. The ‘sideways E’ that you mention is actually the capital greek letter $\boldsymbol{\Sigma}$ (pronounced ‘sigma’). It is used to compactly denote a sum in math, for eg. $\sum_{i=0}^n i = 0+1+2+3+4+…+n$. https://en.wikipedia.org/wiki/Summation
2. The ‘i=0’ and ‘n’ denote the limits of the summation. For eg. if I have $\sum_{i=3}^5 i$ then it will denote the sum $3+4+5$. $i$ is generally assumed to run over natural numbers.
3. The $’i’$ under the x is the index. It is the variable which changes from 0 till n. This basically denotes different values of a specified set, like$x_1, x_2$ and so on. It is just a way to say that all the numbers belong to a list. https://en.wikipedia.org/wiki/Index_notation This must not be confused with the power of a number, which is also sometimes called index.
4. $\overline{x}$ is the average (arithmetic mean) of all the $x_i$s. https://en.wikipedia.org/wiki/Mean#Arithmetic_mean_(AM)