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Statistics

Variance and Standard Deviation

A measure of dispersion is important for statistical analysis. Such concepts find extensive applications in disciplines like finance, business, accounting etc. We rely a lot on such measures from analyzing a stock to studying a student’s performance. We now know about range and mean as measures of dispersion. Now let us study the most widely used tools in the dispersion toolbox- Variance and Standard Deviation.

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Need for Variance and Standard Deviation

We have studied mean deviation as a good measure of dispersion. But a major problem is that mean deviation ignores the signs of deviation, otherwise they would add up to zero. To overcome this limitation variance and standard deviation came into the picture. Unlike mean deviation, standard deviation and variance do not operate on this sort of assumption. Rather they make use of the squares of deviations.

Standard Deviation

Standard deviation is the most important tool for dispersion measurement in a distribution. Technically, the standard deviation is the square root of the arithmetic mean of the squares of deviations of observations from their mean value. It is generally denoted by sigma i.e. σ.

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Note that unlike mean deviation that can be measured by mean, median and mode, S.D. can be measured by mean only. As there are three classes of distributions, we calculate the standard deviation for all three types in a different way. Let us study them individually.

Individual Series

Initially, we calculate the value of the arithmetic mean. Further, we calculate the value of deviation for each observation about mean using the formula: D= X – Mean. In the next step, we divide the summation of squares of these deviations by the number of observations. Finally, the square root of this value is the standard deviation. The formula is as follows:

Standard Deviation (σ)= √[∑D²/N]

Here, D= Deviation of an item relative to mean

N= The number of observations

Discrete Series

In discrete series, each observation is associated with a frequency. Hence, the formula for calculation of standard deviation changes accordingly to include frequency. The formula is as follows:

Standard deviation(σ)= √(∑fD²)/N)

Here, D= Deviation of an item relative to the mean calculated as, D= X – Mean

f= Frequencies corresponding to the observations

N= The Summation of frequency

Frequency Distribution Series

The approach towards the calculation of standard deviation for frequency distribution is pretty much the same as for discrete series. The only difference occurs when using the values of observations. The mid values of the classes are derived dividing the sum of upper and lower value of class and this value is used for calculations. The formula is:

Standard deviation(σ)= √(∑fD²)/N)

Here, D= Deviation of an item relative to the mean calculated as, D= Xi – Mean

f= Frequencies corresponding to the observations

N= The summation of frequency

Another Approach for Standard Deviation

There is another formula for calculation of standard deviation, effectively derived from the traditional formula. The formula is:

Standard Deviation= {√[N∑fx² – ( ∑fx)²]} ÷ N

f = Frequency corresponding to an observation

x= The value of observation (for discrete distribution) or the mid-point of the class (for frequency distribution)

Variance

Variance and Standard deviation

Although standard deviation is the most important tool to measure dispersion, it is essential to know that it is derived from the variance. Variance uses the square of deviations and is better than mean deviation. However, since variance is based on the squares, its unit is the square of the unit of items and mean in the series.

With this in mind, statisticians use the square root of the variance, popularly known as standard deviation. Effectively, the square root of the variance is the standard deviation. With the knowledge of calculating standard deviation, we can easily calculate variance as the square of standard deviation.

Variance = ( Standard deviation)² = σ×σ

Short Method to Calculate Variance and Standard Deviation

We are familiar with a shortcut method for calculation of mean deviation based on the concept of step deviation. Similarly, such a method can also be used to calculate variance and effectively standard deviation. We use this approach when the values of observations (discrete series) or the values of mid-points of intervals (frequency distribution) are too large.

This method simplifies the calculation of mean and variance in such cases. Here, we assume a value in the series as the mean and divide the deviations by width of the intervals ( known as step deviations D’). Lastly, we use the step deviations to calculate standard deviation as follows:

Standard deviation(σ)= √[N×(∑fD’²) – (∑fD’/N)²]  × (h/N)

Here, D’= Step-deviation of observations relative to an assumed value, calculated as D’= (X-A)/h

N= The summation of frequency

h= Width of the class

An Illustration for You

Q: Calculate variance and standard deviation for the following data:

  x 2 4 6 8 10
  f 3 5 9 5 3

Ans:

x f fx D fD²
2 3 6 -4 16 48
4 5 20 -2 4 20
6 9 54 0 0 0
8 5 40 2 4 20
10 3 30 4 16 48
∑= 25 ∑= 150 ∑= 136

Mean= ∑fx/∑f= 150/25= 6

Hence, variance= ∑fD²/N= 136/25= 5.44

And standard deviation= √(5.44)= 2.33

Question: How do you calculate the variance?

Answer: We can define variance as the average of the squared differences from the mean. Besides, for calculating the variance follow these steps:

  • Firstly, work out the mean (simple average of the mean)
  • Next, subtract the mean and square the result for each number.
  • After that, work out the average of those squared differences.

Question: What exactly is a variance?

Answer: Variance in probability theory and statistics is a way to measure how far a set of number is spread out. Moreover, we can describe how much a random variable differs from its expected value.

Question: What is variance derivation?

Answer: Variance which we symbolized by \(S^{2}\) and standard derivation is the most commonly used measures of spread. In addition, we know that the variance is a measure of how to spread out a data set is. Furthermore, it is calculated as the average squared deviation of each number from the mean of the data set.

Question: Why is variance important?

Answer: It is tremendously important as a means to visualize and understand the data being considered. Besides, statistics in a sense were created to represent the data in two or three number.

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