Section 3.3

Measures from Grouped Data

When raw data is unavailable and we only have a frequency distribution, we can still estimate the mean and standard deviation using weighted techniques.

The Weighted Mean

Definition

The Weighted Mean is used when some values contribute more to the average than others. Instead of treating all values equally, each value is multiplied by a weight that reflects its importance or frequency.

Where = weight of each value, = data value

Example: GPA Calculation

A student earns the following grades. Calculate their GPA:

Course	Grade	Points ()	Credits ()
English	A	4	3	12
Statistics	B	3	4	12
History	C	2	3	6
Totals			10	30

InteractiveWeighted Mean Calculator

Formula:

Label	Weight ()
		12.00
		12.00
		6.00
Totals	10.00	30.00

Result:

3.0000Weighted Mean

💡 Tip: For GPA calculations, use grade points (A=4, B=3, C=2, D=1, F=0) as values and credit hours as weights.

Mean from a Frequency Distribution

Why Use Class Midpoints?

When data is grouped into classes, we lose the individual raw values. To estimate the mean, we assume all values within a class are concentrated at the class midpoint (). The midpoint is the average of the lower and upper class limits.

Class Midpoint Formula:

Estimated Mean of Grouped Data:

Where = frequency of class , = class midpoint, and

Example: Estimating Mean from Grouped Data

Class	Frequency ()	Midpoint ()
10 – 19	5	14.5	72.5
20 – 29	12	24.5	294
30 – 39	8	34.5	276
Totals	25	—	642.5

InteractiveGrouped Data Mean Calculator

Formula:

Where = class midpoint, = frequency,

Lower Limit	Midpoint ()	Frequency ()
	14.5		72.50
	24.5		294.00
	34.5		276.00
Totals		n = 25	642.50

Estimated Mean:

25.7000(Estimated Mean)

💡 Note: The midpoint is automatically calculated as . Since we don't know the exact values within each class, this mean is an estimate.

Standard Deviation from Grouped Data

The Sample Standard Deviation Formula

Just like the mean, we can estimate the standard deviation from grouped data by using class midpoints as proxies for the raw data values.

Sample Standard Deviation of Grouped Data:

Where = frequency, = class midpoint, = estimated mean, and

Why These Are Estimates

When data is grouped, we lose the original raw values. We don't know the exact values within each class—only that they fall somewhere in the range. By using the midpoint, we're making an assumption about where the data is centered. This is why calculations from grouped data are always approximations, not exact values.

InteractiveGrouped Data Standard Deviation Calculator

Sample Standard Deviation Formula:

Lower
	14.5		-11.20	125.44	627.20
	24.5		-1.20	1.44	17.28
	34.5		8.80	77.44	619.52
Totals		n = 25			1264.0000

Estimated Mean ()

25.7000

Variance ()

52.6667

Std Deviation ()

7.2572

Calculation:

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