Standard Deviation: Overview, Questions, Preparation

One of the basic topics of Statistics, Standard Deviation, is measuring the spread of data around the mean. If the data points are near to mean, then there is a small deviation, and if they are far from the mean, then there is a high Standard Deviation. 

Standard Deviation is related to another important topic called Variance. A good understanding of both is the base of understanding the data.

Properties of Standard Deviation

• Standard Deviation is also known as the root mean square deviation because Standard Deviation is the square root of averages of squares of all the data in a data set.
• Standard Deviation can never be negative. The value of Standard Deviation is zero or near to it if all the data points in the data are similar. On the other hand, if the data points are far apart, then, the Standard Deviation for the data set is high.

The standard deviation formula helps to measure the amount of dispersion present in the data. In simple terms, it helps to find the amount of deviation in the data from the average. A low value means data points are close to the mean, while a high value indicates that data points are far from the mean. Also, the standard deviation cannot be negative.

The standard deviation formula is used to calculate the standard deviation for a data set. It can be calculated for the whole population or the sample of data. It is necessary to understand these formulas as they will help understand the advanced statistical concepts.

Population Standard Deviation Formula:

The formula for Standard Deviation is as below:
σ = ((Σ (Xi−μ)^2)/N)^1/2

Here, σ is the Standard Deviation.
N is the population size.
Xi is an ith data point in the population, and I varies from 1 to N
μ is the mean of the population

Sample Standard Deviation Formula:

S = ((Σ (Xi− x̄)^2)/n-1)^1/2

Here, Σ is Standard Deviation

μ and x̄ are mean

Xi is the ith data point, where i varies from 1 to n

n is the total data points in the data set

The formula of Standard for Discrete Frequency Distribution
Consider the discrete frequency distribution data as below:
X: x1, x2, x3, …, xn
F: f1, f2, f3, …, fn
The formula is as below:
σ = ((Σ fi(Xi− x̄)^2)/N)^1/2
Here, N is Σ fi where i varies from 1 to n

The formula of Standard Deviation For Grouped Data

σ = ((Σfixi^2 – (Σfixi)^2)^1/2)/N

How to Calculate Standard Deviation?

You can understand the calculation of Standard Deviation with the help of the below points:
Step 1: First of all, find the mean value by adding all the numbers and dividing the sum by the total number of data points.
Step 2: Now, Subtract each data point from the mean calculated in Step 1. Square all the values and add them up. Divide the sum by the total number of data points to get the Variance.
Step 3: Take the square root of the result of Step 2 to get the Standard Deviation.

As part of this topic, students will learn what Standard Deviation is and how it is calculated. Understanding this topic is required to get a better understanding of advanced statistical concepts. Students can expect questions carrying 8 marks from this topic in the exam.

The standard deviation formula is an important topic for class X students. Students are taught about the standard deviation formulas for population, sample, discrete data, and group data. Students can expect at least two numerical questions based on the formulas of the standard deviation of 4-6 marks from this topic.

Calculate the Standard Deviation for the below data: 

Solution: 
Mean = (30+12+5+11+7)/5=13

Variance = ((13-30)^2+(13-12)^2+(13-5)^2+(13-11)^2+(13-7)^2+(13-10)^2)/6
        =((-17)^2+(1)^2+(7)^2+(2)^2+(6)^2)/5
        =(289+1+49+4+36)/5=75.8
Standard Deviation = 75.8^1/2 = 8.71

2. Find the Standard Deviation for below numbers: 
4,9,7,12,13,10,12

Solution: 

Mean (μ) = (4+9+7+12+13+10+12)/7 = 9.57

Standard Deviation = (60.91/7)^½= 8.7^½= 2.95

3. Find the Standard Deviation for the below numbers:
5,3,4,7,8

Solution:

Mean (μ) = (5+3+4+7+8)/6 = 27/6 = 4.5

Standard Deviation = (21.25/5)^1/2 = (4.25)^1/2 = 2.06

4. Below is the data for six students studying average hours in a day. Find the standard deviation for the data.
3,7,6,4,3,4
Solution: Mean = (3+7+6+4+3+4)/6 = 27/6 = 4.6

Standard Deviation = (13.56/6-1)^1/2 = (13.56/5)^1/2 = (2.712)^1/2 = 1.65

5. Find the standard deviation for the weights of eggs given by hen:
65, 61, 66, 67, 56, 59, 70
Solution:

Mean = 65+61+66+67+56+59+70/7 = 63.43

Standard Deviation = SQRT(145.68/7) = SQRT(20.81) = 4.56

6. Find the standard deviation of the daily earnings of a labourer:
55, 65, 65, 67, 69
Solution:

Mean = 54+65+65+67+69/5 = 64

Standard Deviation = SQRT(136/5) = 5.21

Q: What is a Variance?

Q: How is Variance related to the Standard Deviation?

Q: What are the formulas for Standard Deviation and sample Standard Deviation?

Q: Why can the Standard Deviation never have a negative value?

Q: What is the meaning of zero Standard Deviation?

Q: How standard deviation and variance are different?

Q: Define relative standard deviation.

Q: What is the use of standard deviation?

Q: What is the difference between formulas of population and sample standard deviation?

Q: What are different types of distributions available in data?

• Individual Series: It has data in a single column.
• Discrete Series: It has two columns, one for observation and the other for the frequency.
• Frequency Distribution: It has two columns, one with intervals and the other for frequency.