Mean for Grouped Data

When Individual Data Values Are Hidden in Intervals

For ungrouped data, we calculate the mean by adding all values and dividing by the number of values.

In a frequency distribution, however, values may be presented as groups or intervals. Test scores might be grouped into $70\text{-}79$, $80\text{-}89$, and $90\text{-}99$.

Once data is grouped, we no longer know each exact value. If $5$ students belong to the $70\text{-}79$ interval, for example, we cannot tell whether their individual scores were $70$, $72$, $75$, or other values in that interval.

Using the Class Midpoint

Because the exact values are unavailable, we make an approximation: the midpoint of each class represents every value in that interval.

The class midpoint, denoted by $x_i$, is:

Let $L_i$ and $U_i$ be the lower and upper limits of class $i$:

Formula for Mean of Grouped Data

After finding each midpoint, calculate the estimated mean with:

The quantities in this formula are:

• $\bar{x}$ is the estimated mean of the grouped data.
• $f_i$ is the frequency of class $i$.
• $x_i$ is the midpoint of class $i$.
• $\sum (f_i \times x_i)$ is the sum of each class frequency multiplied by its midpoint.
• $\sum f_i$ is the sum of all frequencies, equal to the total number of values $n$.

Average Shoe Size from Grouped Data

Consider the grouped data for shoe sales at Store A:

Calculation:

1. Find the midpoint ($x_i$) of each class:

   • Class $37\text{-}39$: $x_1 = (37+39)/2 = 38$
   • Class $40\text{-}42$: $x_2 = (40+42)/2 = 41$
   • Class $43\text{-}45$: $x_3 = (43+45)/2 = 44$
   • Class $46\text{-}48$: $x_4 = (46+48)/2 = 47$

2. Multiply each frequency by its midpoint ($f_i \times x_i$):

   • Class $37\text{-}39$: $2 \times 38 = 76$
   • Class $40\text{-}42$: $11 \times 41 = 451$
   • Class $43\text{-}45$: $16 \times 44 = 704$
   • Class $46\text{-}48$: $1 \times 47 = 47$

3. Sum all the products ($\sum (f_i \times x_i)$):

   $$\begin{aligned} &\sum (f_i \times x_i) \\ &= 76 + 451 + 704 + 47 \\ &= 1278 \end{aligned}$$

4. Sum all frequencies ($\sum f_i$):

   $$\sum f_i = 2 + 11 + 16 + 1 = 30$$

5. Calculate the mean ($\bar{x}$):

   $$\bar{x} = \frac{\sum (f_i \times x_i)}{\sum f_i} = \frac{1278}{30} = 42.6$$

The estimated average shoe size sold at Store A is therefore $42.6$.

This result is an estimate because the midpoint stands in for every value within its class. It is the standard estimate when only grouped frequencies are available.