Mean for Grouped Data

Mean for Grouped Data

You already know how to calculate the mean for individual data, right? Just sum up all the values and divide by the number of data points.

But what if the data is presented in groups or intervals, like in a frequency distribution table? For example, test scores grouped into 70-79, 80-89, 90-100.

When data is grouped, we don't know the exact value of each data point within that group. For instance, if there are 5 students in the 70-79 group, we don't know if their scores are exactly 70, 72, 75, or some other value within that range.

Using the Class Midpoint

Since we don't know the exact values, we use an assumption or approximation. We assume that all data points within a group are represented by the midpoint of that group (class interval).

The class midpoint (often symbolized as $x_i$) is calculated as:

Formula for Mean of Grouped Data

Once we have the midpoint for each class, we can calculate the mean of the grouped data using the formula:

Where:

• $\bar{x}$ = Mean of the grouped data
• $f_i$ = Frequency of the $i$ -th class (how many data points are in that group)
• $x_i$ = Midpoint of the $i$ -th class
• $\sum (f_i \times x_i)$ = Sum of the products of the frequency and midpoint for each class
• $\sum f_i$ = Sum of all frequencies (equal to the total number of data points, $n$)

Example: Average Shoe Size

Consider the grouped data for shoe sales at Store A:

Steps to calculate the Mean:

1. Find the Midpoint ($x_i$) for each class:

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

2. Multiply Frequency by Midpoint ($f_i \times x_i$) for each class:

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

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

   $$\sum (f_i \times x_i) = 76 + 451 + 704 + 47 = 1278$$

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

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

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

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

Therefore, the average shoe size sold at Store A is 42.6.

Remember, this result is an estimate of the mean because we use midpoints to represent the data within each group. However, this is the standard and most common way to calculate the mean for grouped data.