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To find the variance of grouped data, you need to calculate the mean, standard deviation, and midpoints of the given data set. Then, you can use the formula, Variance = (sum of (midpoint – mean)^2 * frequency)/N, to calculate the variance. To demonstrate, let’s use the data set of exam scores: {40-45: 2, 46-50: 4, 51-55: 5, 56-60: 3}. To find the variance, you first need to calculate the mean, standard deviation, and midpoints of the given data set. The mean is 50.5, the standard deviation is 5.3, and the midpoints are 42.5, 48, 53, and 58. Then, you can use the formula to calculate the variance: (2*(42.5-50.5)^2 + 4*(48-50.5)^2 + 5*(53-50.5)^2 + 3*(58-50.5)^2)/14 = 28.6. Therefore, the variance for the given data set is 28.6.
Often we may want to calculate the of a grouped frequency distribution.
For example, suppose we have the following grouped frequency distribution:
While it’s not possible to calculate the exact variance since we don’t know the , it is possible to estimate the variance using the following formula:
Variance: Σni(mi-μ)2 / (N-1)
where:
- ni: The frequency of the ith group
- mi: The midpoint of the ith group
- μ: The mean
- N: The total sample size
Note: The for each group can be found by taking the average of the lower and upper value in the range. For example, the midpoint for the first group is calculated as: (1+10) / 2 = 5.5.
The following example shows how to use this formula in practice.
Example: Calculate the Variance of Grouped Data
Suppose we have the following grouped data:
Here’s how we would use the formula mentioned earlier to calculate the variance of this grouped data:
We would then calculate the variance as:
- Variance: Σni(mi-μ)2 / (N-1)
- Variance: (604.82 + 382.28 + 68.12 + 477.04 + 511.21) / (23-1)
- Variance: 92.885
The variance of the dataset turns out to be 92.885.
The following tutorials explain how to calculate other metrics for grouped data: