In a grouped data context, which of the following is a common use of midpoints?

In a grouped data context, which of the following is a common use of midpoints?

• A. To estimate the mean
• B. To estimate the median directly
• C. To test for normality
• D. To replace missing values

Answer: (A)

Midpoints are used as representative values for each class interval in grouped data. Because we don’t know the exact values of all observations within a class, we assume every observation in that class equals the class midpoint. This lets us compute the mean by weighting these midpoints by their class frequencies: the estimated mean is the sum of (midpoint × frequency) across all classes, divided by the total number of observations. The mean is a linear statistic, so replacing the unknown values with their class midpoints preserves the overall total well enough to yield a reasonable estimate. For the median, you’d typically use the cumulative frequencies and interpolate within the median class to approximate its location, rather than relying on midpoints alone. Normality tests assess the shape of the entire distribution rather than rely on a single representative value per class, and while missing values can be imputed, midpoints aren’t the standard or primary method to replace them in grouped data.

Midpoints are used as representative values for each class interval in grouped data. Because we don’t know the exact values of all observations within a class, we assume every observation in that class equals the class midpoint. This lets us compute the mean by weighting these midpoints by their class frequencies: the estimated mean is the sum of (midpoint × frequency) across all classes, divided by the total number of observations. The mean is a linear statistic, so replacing the unknown values with their class midpoints preserves the overall total well enough to yield a reasonable estimate.

For the median, you’d typically use the cumulative frequencies and interpolate within the median class to approximate its location, rather than relying on midpoints alone. Normality tests assess the shape of the entire distribution rather than rely on a single representative value per class, and while missing values can be imputed, midpoints aren’t the standard or primary method to replace them in grouped data.