Mathematics

Standard deviation and variance

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In this article we shall explore two important measures of dispersion in statistics. The standard deviation and variance for discrete data. Below is a list of notes covered in this article.

Quick summary

  • The deviation of an observation x from the mean is given by x – x.
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  • We can work out the variance and standard deviation for a frequency table and a grouped frequency distribution by using x as the mid-point of the class, and n = Σf. Then; [IMAGE]
  • Coding can be used to simplify data with very large values. To find the standard deviation of original data, find the standard deviation of the coded data and either multiply this with what you divided by or divide this by what you multiplied by.
Variance: Is the total dispersion (spread) of data from the mean.

We can find the deviation of a single observation x from the mean by;
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The variance is the total of the deviations from the mean for all the observations in a data set.
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Another version of the variance formula is shown below;
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Notice that variance is measured in units2. The square root of variance is called the Standard Deviation
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Finding Variance and Standard Deviation

Example

The marks scored in a test by random selected students in a class are shown below;
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  • Find the variance of the marks
Answer

We first need to find the mean;
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Now we can find the deviation from the mean for each observation. Deviation = x – x for example for the observation 4.
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For the rest of the data we get the following deviation;
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Now we will need to square each observation;
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To find the variance we find the total for (deviation)² and divide by n;
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Example

Using the same data in the previous question
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Use an alternative formula to find the variance and standard deviation of the marks.

Answer

The following formula is much quicker and easier to use than the formula used in the previous example;
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We need only find x², total of x² and total of x.
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Frequency tables and grouped frequency distribution

When carrying out calculations of variance and standard deviation for a frequency distribution we must treat x as the mid-point of the class and n should stand for Σf. Therefore the formula to find variance;
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Example
In a very large company Grace records the time spent by employees during the lunch hour to the nearest minute, x, out of the company building. The results are as follows.
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  • Calculate the standard deviation of the time spent out of the building
Answer
To make calculations easier we shall add two columns to the table, fx and fx². We shall also add a total row at the bottom of the table;
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We can therefore use the values in the variance formula;
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Whenever you use a mid-point of an interval to find the variance and standard deviation, the answer is an estimate.

Example

Andrew recorded the lenth in minutes of each telephone call he made in one month. The results are shown in the table below.
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  • Calculate an estimate of the standard deviation of the length of telephone calls.
Answer

In the question we will need to use the mid-point as x. To make the calculations easier we shall add 1 row and three more columns to the table
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Coding

When working with variance and standard deviation we can use the coding to make numbers easier to work with when the data involves large values. Below is a quick summary of what you should know.

Summary

  • Adding or subtracting numbers does not change the standard deviation of data
  • Multiplication or division of the data does not affect the standard deviation.
  • To find the standard deviation of the original data, find the standard deviation of the coded data and either multiply this by what you divided the observations by or divide this with what you multiplied the observations by.
Example

  1. Find the standard deviation of the following lengths
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  2. Use the given coding to find the standard deviation of the above data
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Answers

  1. To find the standard deviation we must first find the variance and square root it.
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    The coding is y=x/10. Therefore the standard deviation of the coded data is the standard deviation of the original data divided by 10.

    Standard deviation of coded data

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    To find the standard deviation of the original data multiply the standard deviation of the coded data by 10.
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    The coding we have to use is;
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    Adding or subtracting a number does not change the spread of the data.

    Notice that the range for the coded data is still the same as the range for the original data. Therefore the standard deviation is the same for the coded data as for the original data.

    Standard deviation of coded data

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    The standard deviation in this answer is the same as the standard deviation for the original answer.

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    Adding or subtracting a number does not change the spread of the data.

    However the division by 10 will make a difference.
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    Standard deviation of coded data is 1.41

Example
Data is coded by using;
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The standard deviation of the coded data is 2.5

  • Find the standard deviation of the original data.
Answer

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