Variance and Standard Deviation for Grouped Data
You may be faced with finding the variance and standard deviation of data that has been collected and organized into a frequency distribution table and may not have access to the original data.
Here’s how we work with the data in the distribution table.
Here’s an example of organized data.
The amount of caffeine in a sample of 5ounce servings of brewed coffee is given in the distribution table. Use this data to estimate the samples mean, variance, and standard deviation.
Classes 
Frequency 
60 – 81 
1 
82 – 103 
12 
104 – 125 
25 
126 – 147 
10 
148 – 169 
2 
Use the midpoint of each class to represent the entire class. Assign weight.
Classes 
Frequency 
Midpoint, x 
f ^{.} x 
60 – 81 
1 
70.5 
70.5 
82 – 103 
12 
92.5 
1110 
104 – 125 
25 
114.5 
2862.5 
126 – 147 
10 
136.5 
1365 
148 – 169 
2 
158.5 
317 
Classes 
Frequency 
Midpoint, x 

60 – 81 
1 
70.5 
44 
82 – 103 
12 
92.5 
22 
104 – 125 
25 
114.5 
0 
126 – 147 
10 
136.5 
22 
148 – 169 
2 
158.5 
44 
.
Next, find the square of the deviations and assign weight.
Classes 
Frequency 
Midpoint, x 

60 – 81 
1 
70.5 
44 
1936 
1936 
82 – 103 
12 
92.5 
22 
484 
5808 
104 – 125 
25 
114.5 
0 
0 
0 
126 – 147 
10 
136.5 
22 
484 
4840 
148 – 169 
2 
158.5 
44 
1936 
3872 