Standard Deviation of Grouped Data

Date: 10/16/2000 at 16:00:33
From: Laura
Subject: Standard deviation

Interval (grouped) data:

     Interval (group)     Frequency
     ----------------    -----------
          37-46              19
          47-56              23
          57-66              27
          67-76              28

What is the standard deviation of the data?

I have another question. If I had an answer of 31.615 for the mean of 
a different data set, would I round it off to 32 or 31.62?

Date: 10/17/2000 at 13:59:27
From: Doctor TWE
Subject: Re: Standard deviation

Hi Laura - thanks for writing to Dr. Math.

I'll break this down by steps.

Step 1: Find the number of data points.
To find the number of data points, add up the values in the Frequency 
column of the table:

     Interval     Freq.
     --------     -----
      37-46        19
      47-56        23
      57-66        27
      67-76        28
                  ----
                   97

Step 2: Find the midpoint of each interval range.
To find the midpoint, add the top and bottom of each interval range 
and divide by two. For example, the first interval range is 37 to 46, 
so the midpoint is:

     (37 + 46) / 2  =  83 / 2 = 41.5

Do this for each interval range. Add a column to your table for this 
(I'll put it between the Interval and Frequency columns):

     Interval   Midpt.   Freq.
     --------   ------   -----
      37-46      41.5     19
      47-56       :       23
      57-66       :       27
      67-76       :       28
                         ----
                          97

Step 3: Find the estimated sum of the data.
To find the sum, multiply the midpoint of each interval range by the 
frequency of that interval range. For example, the midpoint of the 
first interval range is 41.5 and the frequency is 19, so the sum is:

     41.5 * 19 = 788.5

Do this for each interval range. Add another column to your table for 
this (I'll put it after the Frequency column), then find the sum of 
that column (I'll just call this S):

     Interval   Midpt.   Freq.    Sum
     --------   ------   -----   -----
      37-46      41.5     19     788.5
      47-56       :       23       :
      57-66       :       27       :
      67-76       :       28       :
                         ----    -----
                          97       S

Step 4: Find the estimated mean (or "average") of the data.
Divide the sum of the data (S, found in step 3) by the number of data 
points (found in step 1). In our example,

     Mean = S / 97

Step 5: Find the squares of the midpoints of each interval range.
For each interval range, find the square of the midpoint. Add another 
column to your table for this (I'll put it after the Sum column). For 
example, the midpoint of the first interval range is 41.5, so the 
square is:

     41.5^2 = 1722.25

Do this for each interval range:

     Interval   Midpt.   Freq.    Sum    Midpt^2
     --------   ------   -----   -----   -------
      37-46      41.5     19     788.5   1722.25
      47-56       :       23       :        :
      57-66       :       27       :        :
      67-76       :       28       :        :
                         ----    ----- 
                          97       S        

Step 6: Find the estimated sum-of-the-squares of the data.
To find the sum-of-the-squares, multiply the square of the midpoint of 
each interval range by the frequency of that interval range. For 
example, the square of the midpoint of the first interval range is 
1722.25 and the frequency is 19, so the sum-of-the-squares is:

     1722.25 * 19 = 32722.75

Do this for each interval range. Add another column to your table for 
this (I'll put it after the Midpt^2 column), then find the sum of that 
column (I'll just call this S2):

     Interval   Midpt.   Freq.    Sum    Midpt^2   Sum-Sqrs
     --------   ------   -----   -----   -------   --------
      37-46      41.5     19     788.5   1722.25   32722.75
      47-56       :       23       :        :         :
      57-66       :       27       :        :         :
      67-76       :       28       :        :         :
                         ----    -----             --------
                          97       S                  S2

Step 7: Find the estimated mean square of the data.
Divide the sum-of-the-squares of the data (S2, found in step 6) by the 
number of data points (found in step 1). In our example,

     Mean square = S2 / 97

Step 8: Find the estimated variance and standard deviation of the 
data. 
To find the variance, square the mean (from step 4), then subtract it 
from the mean square. Note that the mean square and the square of the 
mean are not the same!

     Var = (Mean square) - (Mean)^2

To find the standard deviation, take the square root of the variance.

     StDev = sqrt(Var)

Note that these values are estimates, because with grouped data, you 
don't have the exact figures to work with. Your means, squares, 
variance and standard deviation are all based on estimations of the 
actual data.


>I have another question. If I had an answer of 31.615 for the mean of 
>a different data set, would I round it off to 32 or 31.62?

That depends on the accuracy and precision of the original data. In 
some scientific fields, there are very specific rules for determining 
the number of significant figures to leave in an answer, and they can 
get quite complicated. As a general rule, your final answer should 
have the same precision (i.e. the same number of decimal places) as 
the LEAST precise data point. So, for example, if I had the data set:

     16.725, 31.0625, 24.5, 22.50, 19.75

I'd compute the mean as:

     (16.725 + 31.0625 + 24.5 + 22.50 + 19.75) / 5 = 22.9075

Then I'd round it to 22.9 (NOT 22.91) because my least precise data 
point (the 24.5) had only one decimal place in it.

I hope this helps. If you have any more questions, write back.

- Doctor TWE, The Math Forum
  http://mathforum.org/dr.math/