To approximate the mean from a grouped frequency distribution table (GFDT), we use the formula:

$\text{Mean} = \frac{\sum (f \times x)}{\sum f}$

where:

• $f$ is the frequency for each class interval,
• $x$ is the midpoint of each class interval.

Step 1: Calculate the midpoints ($x$) for each class interval.

Midpoint for a class interval is calculated as:

$x = \frac{\text{Lower Bound} + \text{Upper Bound}}{2}$

Let's calculate the midpoints for each class interval:

Data	Midpoint ($x$)	Frequency ($f$)
70 - 74	$\frac{70 + 74}{2} = 72$	1
75 - 79	$\frac{75 + 79}{2} = 77$	3
80 - 84	$\frac{80 + 84}{2} = 82$	8
85 - 89	$\frac{85 + 89}{2} = 87$	12
90 - 94	$\frac{90 + 94}{2} = 92$	17
95 - 99	$\frac{95 + 99}{2} = 97$	15
100 - 104	$\frac{100 + 104}{2} = 102$	9
105 - 109	$\frac{105 + 109}{2} = 107$	5
110 - 114	$\frac{110 + 114}{2} = 112$	1

Step 2: Multiply frequency by midpoint ($f \times x$) for each class interval.

Now calculate $f \times x$ for each interval:

\hline \text{Data} & \text{Midpoint (x)} & \text{Frequency (f)} & f \times x \\ \hline 70 - 74 & 72 & 1 & 72 \\ 75 - 79 & 77 & 3 & 231 \\ 80 - 84 & 82 & 8 & 656 \\ 85 - 89 & 87 & 12 & 1044 \\ 90 - 94 & 92 & 17 & 1564 \\ 95 - 99 & 97 & 15 & 1455 \\ 100 - 104 & 102 & 9 & 918 \\ 105 - 109 & 107 & 5 & 535 \\ 110 - 114 & 112 & 1 & 112 \\ \hline \end{array}$$ ### Step 3: Calculate the sum of $$f \times x$$ and the sum of $$f$$. $$\sum (f \times x) = 72 + 231 + 656 + 1044 + 1564 + 1455 + 918 + 535 + 112 = 6587$$ $$\sum f = 1 + 3 + 8 + 12 + 17 + 15 + 9 + 5 + 1 = 71$$ ### Step 4: Calculate the mean. $$\text{Mean} = \frac{6587}{71} \approx 92.8$$ Thus, the approximate mean is **92.8**. Would you like more details or have any questions? --- **Related questions:** 1. What is the standard deviation for this dataset? 2. How would the median of this dataset be calculated? 3. What is the mode of this grouped frequency distribution? 4. Can we calculate the range based on this GFDT? 5. How does the class width affect the accuracy of the mean approximation? **Tip:** The accuracy of a mean approximation depends on the number of class intervals and how well the midpoints represent the data points in each class.