Q 605. What is the pooled standard deviation? What are the different methods of calculating it and how does it affect the final value (provide examples to support your answer)

 

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The pooled standard deviation is  considered especially in the case - if we have two or more than two groups and we also  have the condition that the standard deviations of both  groups are approximately the same, then we try to take  the standard weighted average deviations. of them. groups 
 Now if we consider the two groups as group1 and group2, where 
 
 Group 1 Group 2 
 Sample size (n1): 15 Sample size (n2): 19 
 Standard deviation (s1) : 6.4 Standard deviation (s2) : 8.2 
 
 Now that we know the  above formulas to calculate the PSD of these two groups 
 Sum standard deviation = √ (15-1) 6.42 (19-1) 8.22 / (15 19-2) = 7.466 
 Note how the pooled standard deviation (7.466) is between the Group 1 (6.4) and Group 2 (8.2) values. 
 This indicates that the population standard deviation is specifically the weighted average between the two groups.

Steps to calculate the pooled standard deviation 
 1.Calculate the difference between each value and the means of that group. 
 2. Square those differences. 
 3. Add them all together (for all groups). 
 4.Divide by the number of degrees of freedom (total sample size minus the number of groups). 
 5. Take the square root of the last number.

 
 

Pooled standard deviation is the weighted average standard deviation for two or more independent groups. Larger samples are given more weight in decision. There are two method of calculating pooled-standard deviation:

1.

2.

 

2nd method is supportive only when sample size is same for individual groups. lets us understand impact of both methods with the help of examples:

Suppose we have four individual groups of solar cell efficiency with same sample sizes ( shown in below table). Pooled standard deviation value will be “ 0.061826” by using first method . If we use second method then pooled standard deviation value will also be “0.061826” . Both methods are showing same pooled standard deviation value because 1st and 2nd methods will perform same in case of same sample sizes.

 

 

To discern between both methods, let's change sample size of Group-1 from value "6" to "5" and let's check pooled standard deviation for both methods. Now it shows different pooled standard deviation due to sample size non-equal for individual groups. In this case pooled-std deviation value " 0.0634" is acceptable. it indicates that "more weight given to larger samples".

The pooled standard deviation is a weighted average of the Standard deviation from two or more groups of data. It is process for estimating variance of several groups when the mean of each group is different. But it should be assumed that variance of each group is same.

It is also known as Combined standard deviation or Composite standard deviation or Overall standard deviation.

If the sample size is larger, the proportional effect will be greater on the overall estimate of the standard deviation.

Formula for calculating pooled standard deviation for Three of more groups,

Where,

n₁, n₂ – The sample size of the group1, group2

S₁, S₂ – Standard deviation of the group1, group2

k – The number of groups

Formula for Equal sample sizes,

 

Assumptions:

-        Standard deviations of groups are assumed to be equal.

-        It gives weight to the group with larger sample size.

 

Example with different sample sizes:

Consider there are two groups as mentioned below,

 

So the Standard deviation for group 1 is 6.4 and group 2 is 8.2 and from the calculation the Pooled standard deviation for this 2 groups is 7.46

The value of pooled standard deviation 7.46 is within the value of Standard deviation s1 and s2. By this it was understood that pooled standard deviation is a weighted average of groups.     

Application:

-        Ability to compare outcomes and validate across models. like Railway ticket reservation among various stations (PQWL – Pooled Quota Waiting list)

-        Health care

-        Sales Forecasting

-        2-Sample t-test and ANOVA studies.

Pooled standard deviation is a way of calculating the variance of multiple populations with different means but same variance. Formula for measuring pooled SD is given below:
 

1. Normal
 

2. Cohen's formula

 

3. For three or more groups

 

 

4. For equal sample size

 

Ashutosh Bhardwaj has given the best answer to this question. He has clearly explained the different methods of calculating Pooled Standard Deviations and the impact of different calculations (of course depending on the situation).

 

Answer from Keerthi is also a must read.