Calculating Spread for Grouped Data
A grouped frequency table can show how often phone-battery durations fall within intervals such as and . We still want to describe how widely those durations vary.
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The table no longer contains each exact observation. Within , for example, we cannot tell whether a battery lasted , , or another duration.
To estimate the spread, we represent every observation in a class by that class's midpoint . This is a modeling approximation, so the resulting variance and standard deviation estimate the values we would obtain from the ungrouped observations.
We weight each midpoint by its class frequency . The formulas below treat the table as the complete population. A sample requires the corresponding sample-variance formula.
Variance ()
For grouped data, use the following computational formula:
The first term is the frequency-weighted mean of the squared midpoints. The second is the square of the frequency-weighted mean. Their difference is the estimated population variance.
Standard Deviation ()
Just like with ungrouped data, the standard deviation is the square root of the variance:
Suppose a study records phone-battery duration in the following intervals:
| Battery duration (hours) | Frequency () |
|---|---|
We will estimate the population variance and standard deviation of these durations.
For each class, calculate the midpoint , then calculate and .
| Battery duration (hours) | Midpoint, | Frequency, | ||
|---|---|---|---|---|
Plug the total values from the table into the variance formula:
The estimated variance is square hours. Variance uses squared units because every distance from the mean is squared before averaging.
Take the square root of the variance:
The estimated standard deviation is . Around the estimated mean of , durations differ by about in the root-mean-square sense.
When datasets use the same unit and a comparable scale, a smaller standard deviation indicates that values are clustered more tightly around their mean.
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