Variance & Standard Deviation For Ungrouped Data | Definition, Solved Examples, Important Questions, FAQs

Variance & Standard Deviation for Ungrouped and Grouped Data – Class 11 Statistics Notes

Class 11 Mathematics | Written by Neeraj Anand

Published by ANAND TECHNICAL PUBLISHERS

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Introduction to Variance and Standard Deviation

Variance and standard deviation are essential statistical measures that describe the spread or dispersion of a data set. These measures help in understanding how much individual data points differ from the mean.

• Variance (σ²) quantifies the degree of spread in data.
• Standard deviation (σ) is the square root of variance and gives a more interpretable measure of dispersion.

Variance and Standard Deviation are the two important measurements in statistics. Variance is a measure of how data points vary from the mean, whereas standard deviation is the measure of the distribution of statistical data. The basic difference between both is standard deviation is represented in the same units as the mean of data, while the variance is represented in squared units.

Variance

Variance is defined as, “The measure of how far the set of data is dispersed from their mean value”. Variance is represented with the symbol σ². In other words, we can also say that the variance is the average of the squared difference from the mean.

Variance and Standard Deviation Formula

As discussed, the variance of the data set is the average square distance between the mean value and each data value. And standard deviation defines the spread of data values around the mean.

Properties of Variance

Various properties of the Variance of the group of data are,

• As each term in the variance formula is firstly squared and then their mean is found, it is always a non-negative value, i.e. mean can be either positive or can be zero but it can never be negative.
• Variance is always measured in squared units. For example, if we have to find the variance of the height of the student in a class, and if the height of the student is given in cm then the variance is calculated in cm². 

Variance Formula

There are two formulas for Variance, that are:

• Population Variance
• Sample Variance

The formulas for the variance and the standard deviation for both population and sample data set are given below:

Formula for Population Variance

The population variance formula is given by:

\(\begin{array}{l}\sigma^2 =\frac{1}{N}\sum_{i=1}^{N}(X_i-\mu)^2\end{array} \)

Here,

σ² = Population variance

N = Number of observations in population

X_i = ith observation in the population

μ = Population mean

Formula for Sample Variance

The sample variance formula is given as:

\(\begin{array}{l}s^2 =\frac{1}{n-1}\sum_{i=1}^{n}(x_i-\overline{x})^2\end{array} \)

Here,

s² = Sample variance

n = Number of observations in sample

x_i = ith observation in the sample

\(\begin{array}{l}\overline x\end{array} \) = Sample mean

Standard Deviation

Standard deviation is defined as the “spread of the statistical data from the mean or average position”. We denote the standard deviation of the data using the symbol σ.

We can also define the standard deviation as the square root of the variance.

Properties of Standard Deviation

Various properties of the Variance of the group of data are,

• Standard Deviation is the square root of the variance of the given data set. It is also called root mean square deviation. 
• Standard Deviation is a non-negative quantity i.e. it always has positive values or zero values.
• If all the values in a data set are similar then Standard Deviation has a value close to zero. Whereas if the values in a data set are very different from each other then standard deviation has a high positive value.

Standard Deviation Formula

There are two formulas for the standard deviation listed as follows:

• Population Standard Deviation
• Sample Standard Deviation

Formula for Population Standard Deviation

The population standard deviation formula is given as:

\(\begin{array}{l}\sigma =\sqrt{\frac{1}{N}\sum_{i=1}^{N}(X_i-\mu)^2}\end{array} \)

Here,

σ = Population standard deviation

Formula for Sample Standard Deviation

Similarly, the sample standard deviation formula is:

\(\begin{array}{l}s =\sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(x_i-\overline{x})^2}\end{array} \)

Here,

s = Sample standard deviation

Relation between Standard Deviation and Variance

Variance and Standard deviation are the most common measure of the given set of data. They are used to find the deviation of the values from their mean value or the spread of all the values of the data set.

• Variance is defined as the average degree through which all the values of a given data set deviate from the mean value.
• Standard Deviation is the degree to which the values in a data set are spread out with respect to the mean value.

The relationship between Variance and Standard Deviation is discussed below.

Variance = (Standard Deviation)²

OR

√(Variance) = Standard Deviation

Differences Between Standard Deviation and Variance

The differences between Standard Deviation and Variance are discussed in the table below,

Standard Deviation	Variance
Standard Deviation is defined as the square root of  the variance.	Variance is defined as the average of the squared  differences from the mean.
Standard deviation provides a measure of the  typical distance between data points and the mean.	Variance provides a measure of the average  squared distance between data points and the mean.
It is represented by the Greek symbol σ.	It is represented by a square of the Greek symbol sigma  i.e. σ2.
It has the same unit as the data set.	Its unit is the square of the unit of the data set.
It represents the volatility in the market or given data set.	It represents the degree to which the average return  varies according to the long-term change in the market.

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Solved Examples on Variance and Standard Deviation

Example 1: Find the variance and standard deviation of all the possibilities of rolling a die.

Solution:

All possible outcomes of rolling a die are {1; 2; 3; 4; 5; 6}.

This data set has six values (n) = 6

Before finding the variance, we need to find the mean of the data set.

Mean, x̅ = (1+2+3+4+5+6)/6 = 3.5

We can put the value of data and mean in the formula to get;

σ² = Σ (x_i – x̅)²/n

⇒ σ² = [(1-3.5)² + (2-3.5)² + (3-3.5)² + (4-3.5)² + (5-3.5)² + (6-3.5)²]/6

⇒ σ² = (6.25+2.25+0.25+0.25+2.25+6.25)/6

Variance (σ²) = 2.917

Now,

Standard Deviation (σ) = √ (σ²)

⇒ Standard Deviation (σ)  = √(2.917) 

⇒ Standard Deviation (σ) = 1.708

Example 2: Find the variance and standard deviation of all the even numbers less than 10.

Solution:

Even Numbers less than 10 are {0, 2, 4, 6, 8}

This data set has five values (n) = 5

Before finding the variance, we need to find the mean of the data set.

Mean, x̅ = (0+2+4+6+8)/5 = 4

We can put the value of data and mean in the formula to get;

σ² = Σ (x_i – x̅)²/n

⇒ σ² = [(0-4)² + (2-4)² + (4-4)² + (6-4)² + (8-4)²]/5

⇒ σ² = (16 + 4 + 0 + 4 + 16)/5 = 40/5

Variance (σ²) = 8

Now, Standard Deviation (σ) = √ (σ²)

⇒ Standard Deviation (σ)  = √(8) 

⇒ Standard Deviation (σ) = 2.828

Practice Problems – Variance and Standard Deviation

Try solving the given question to grab the idea of what we just learned :

Question 1 : Find the variance and standard deviation of the following data

5

8

3

4

7

12

5

2

Question 2 : Find the standard deviation of the following data:

X	5	10	15	20	25	30
f(x)	6	7	3	2	1	1

Question 3 : Find the variance of the first 69 natural numbers.

Answer 1 : Mean x̄ = 6

Population Variance = 8.5

Standard Deviation = √(8.5) = 2.91

Answer 2 : Mean x̄ = 12

Population Variance = 48.5

Thus, Standard Deviation = √48.5 = 6.96

Answer 3 : Variance = 396.67

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FAQs on Variance and Standard Deviation

What Does Variance Mean?

Variance is defined as the statistical measurement which determines how far each number is from the mean value in a given data set. It tells us about the degree to which our average value can deviate in the data set.

What Does Standard Deviation Mean?

Standard deviation is the measure of the volatility from the mean value in the given data set. It is calculated by taking the square root of the variance of the given data set. It calculates the risk related to the change in values in the market.

What Is the Difference Between Standard Deviation and Variance?

Variance is calculated by taking the average of the squared deviation from the mean, whereas standard deviation is the square root of the variance. The other difference between them is in their unit. Standard deviation is expressed in the same units as the original values while Variance is expressed in unit².

What is Variance formula?

Formula to calculate the variance of the given data set is,

σ² = Σ (x_i – x̅)²/n

What is Standard Deviation formula?

Formula to calculate the variance of the given data set is,

σ = √(Σ (x_i – x̅)²/n)

Does standard deviation have units?

As standard deviation tells us about the volatility of the data thus It has the same unit as the data in the given data set.

Key Properties of Variance and Standard Deviation

1. Variance is always non-negative since it involves squared differences.
2. Standard deviation is expressed in the same unit as the original data, making it easier to interpret.
3. A larger variance or standard deviation indicates more spread in the data.
4. If all data points are the same, the standard deviation is zero.

Conclusion

Variance and standard deviation are crucial measures in statistics, helping us understand data variability. The formulas differ slightly for ungrouped and grouped data, but both follow a fundamental principle of measuring the deviation from the mean.

Students preparing for CBSE Class 11, JEE Mains, and Advanced should practice these concepts thoroughly.

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