Standard Deviation - GMAT Math Study Guide

Table of Contents

  1. Definitions
  2. What is Standard Deviation?
  3. Calculating Standard Deviation
  4. A Graphical Understanding
  5. Testing of Standard Deviation

Definitions

  • Standard Deviation - A measure of how spread out or dispersed the data in a set are relative to the set's mean.
    For example, a data set with a standard deviation of 10 is more spread out than a data set with a standard deviation of 5.

What is the Standard Deviation?

The standard deviation is a measure of the dispersion (i.e., the degree to which data are spread out) relative to the mean (i.e., arithmetic mean).

Although you will likely not need to calculate the standard deviation as it is a tedious mathematical process that does not test competency, many students find seeing and comprehending the standard deviation formula helpful in gaining a deeper understanding of exactly what the standard deviation is and what is measures.

standard deviation formula
standard deviation variables

Calculating Standard Deviation

The steps to calculating the standard deviation roughly follow the above formula.

Set S: 2, 5, 9, 12, 15, 17
  1. Calculate μ, the mean of the set: 60/6 = 10
  2. Find the difference between each term and the mean
    xi (xi - μ) (xi - μ)2
    2 (2-10) (2-10)2 = 64
    5 (5-10) (5-10)2 = 25
    9 (9-10) (9-10)2 = 1
    12 (12-10) (12-10)2 = 4
    15 (15-10) (15-10)2 = 25
    17 (17-10) (17-10)2 = 49
  3. Calculate the sum of the differences between each value and the mean squared:
    64 + 25 + 1 + 4 + 25 + 49 = 168
  4. Divide the previous sum by the number of terms:
    168/6 = 28
  5. Take the square root of the previous value:
    281/2 = 5.2915026
A Few Notes for Advanced Students:
(1) 281/2 is the square root of 28 [see radicals]
(2) The formula above is for the population standard deviation, denoted σ, and is therefore based upon the population mean, denoted μ
(2a) The sample standard deviation, denoted S, is based upon the sample mean, denoted x and not the population mean (i.e., μ). The sample standard deviation is calculated by dividing the sum of (xi - x)2 by n-1 and not n.

A Graphical Understanding

Although the chances of being asked to calculate the standard deviation are virtually zero, it is important to understand the concept that the standard deviation captures. The crux of standard deviation as a tool of descriptive statistics is its ability to measure the extent to which data is dispersed. As the data in a set become more dispersed (i.e., data points tend to exist farther from the mean), the standard deviation increases.

The concept of the dispersion of data, captured in the standard deviation, is visible in graphs of data. The following data sets each have the following characteristics:

standard deviation graph

standard deviation key

(To interact with this graph, please download the underlying Excel file.)

Testing of Standard Deviation

The testing of standard deviation tends to be more conceptual than empirical.

If 25 students took a test, the average student scored 80% (arithmetic mean), the standard deviation was 5%, and the teacher awards As for those who scored 3 or more standard deviations above the mean, what is the lowest score for which a student could earn an A?

μ = 80% since this is arithmetic mean.
Lowest Score to Earn an A: Mean + 3(Standard Deviation)
Lowest Score to Earn an A: 80% + 3(5%) = 95%