What is the median and standard deviation of a distribution are 50 and 5 respectively, if each item is increased by 4.

Median will go up by 2 but S.D. will remain same

Median will increase and S.D. will increase

Median will increase and S.D. will decrease

Exams Academic Standard Deviation And Variance: Definition

Written By Vaibhav_Raj_Asthana
Last Modified 25-01-2023

Standard Deviation And Variance: Definition

Standard Deviation And Variance: As per the definition of Standard Deviation in NCERT “the proper measure of dispersion about the mean of a set of observations is expressed as positive square-root of the variance and is called standard deviation. Therefore, the standard deviation, usually denoted by σ “. The formula for standard deviation is:

\(\sigma=\sqrt{\frac{1}{N}\sum_{i=1}^{n} (x_i-\bar x)^2}\)

Here we will provide you with all the details regarding Standard deviation such as the formula for ungrouped data, frequency distribution (discrete), frequency distribution (continuous), sample questions and other relevant information.

Standard Deviation & Variance Formula

Mean Deviation, Variance, and Standard Deviation are topics from the Statistics chapter of Class 11 Maths. We have provided the formula for all 3 below based on the type of data set one is referring to:

Mean Deviation for the ungrouped data:
- (i) \(M.D.(\bar x)=\frac{\sum \left | x_i-\bar x \right |}{n}\)
- (ii) \(M.D.(M)=\frac{\sum \left | x_i-M \right |}{n}\)
Mean Deviation for the grouped data:
- (i) \(M.D.(\bar x)=\frac{\sum f_i|x_i-\bar x|}{N}\)
- (ii) \(M.D.(M)=\frac{\sum f_i|x_i-M|}{N}\)
Variance and Standard Deviation for the ungrouped data:
- (i) \(\sigma ^2=\frac{1}{N}\sum (x_i-\bar x)^2\)
- (ii) \(\sigma=\sqrt{\frac{1}{N}\sum (x_i-\bar x)^2}\)
Variance and Standard Deviation of a frequency distribution (discrete):
- (i) \(\sigma ^2=\frac{1}{N}\sum f_i(x_i-\bar x)^2\)
- (ii) \(\sigma=\sqrt{\frac{1}{N}\sum f_i(x_i-\bar x)^2}\)
Variance and Standard Deviation of a frequency distribution (continuous):
- (i) \(\sigma ^2=\frac{1}{N}\sum f_i(x_i-\bar x)^2\)
- (ii) \(\sigma=\frac{1}{N}\sqrt{N\sum f_ix_i^2-(\sum f_ix_i)^2}\)

The above formula will help you in your preparation and you can use these to prepare a formula sheet for revision purposes.

Standard Deviation Calculation

Here we will explain the step-by-step process to calculate the standard deviation along with calculating mean and variance.

-1st Step: Consider the below example:

-2nd Step: From the given data, we construct a table

-3rd Step: Calculate the mean, variance, and standard deviation using the formulas learnt.

Standard Deviation Solved Example

Here is a solved example where you have to calculate only the SD (σ).

Example 1. Find the standard deviation for the following data:
\(x_{i}\) – 3, 8, 13, 18, 23
\(f_{i}\) – 7, 10, 15, 10, 6

Solution. Let us create a table for different values.

σ = \(\frac{1}{48}\) x 293.77 => 6.12.

Frequently Asked Questions – FAQs

Here are some questions that are mostly asked on the topic.

Q. How do you calculate the standard deviation?
Ans. To calculate standard deviation follow the steps:
Step 1: Find the mean for the given data set.
Step 2: Calculate the variance (the formula is given on this page with a solved example).
Step 3: Now, take the square root of the variance to find the value of σ.

Q. What does Standard Deviation tell you?
Ans. σ or SD tells us how far is the mean from each observation.

Q. What is the standard deviation and variance?
Ans. Variance estimates the average degree to which each observation differs from the mean and SD is the sqrt of variance.

Q. What is the standard deviation example?
Ans. Let say a data set is given with values 2, 1, 3, 2, 4. The mean and the sum of squares of deviations of the observations from the mean will be 2.4 and 5.2, respectively. So, σ will be √(5.2/5) = 1.01.

Q. What is the use of standard deviation?
Ans. It is used to estimate how a group of observations (i.e., data set) are spread out from the mean (average or expected value).

Learn Standard Deviation Concepts via Video Here

That was all on Standard Deviation, we hope the information provided by us was helpful. However, if you have further questions feel to use the comments section and we will provide you with an update.