UpStudy Free Solution:

To approximate the mean from a grouped frequency distribution table (GFDT), follow these steps:

1. Find the midpoint for each class interval.

2. Multiply each midpoint by the corresponding frequency.

3. Sum these products.

4. Divide the sum by the total frequency.

Let's perform these steps for the given data:

Step 1: Calculate Midpoints (Class Marks)

The midpoint for each class interval is calculated as follows:

\[\text { Midpoint} = \frac { \text { Lower limit} + \text { Upper limit} } { 2} \]

- For \(70 - 74\): \(\frac { 70 + 74} { 2} = 72\)

- For \(75 - 79\): \(\frac { 75 + 79} { 2} = 77\)

- For \(80 - 84\): \(\frac { 80 + 84} { 2} = 82\)

- For \(85 - 89\): \(\frac { 85 + 89} { 2} = 87\)

- For \(90 - 94\): \(\frac { 90 + 94} { 2} = 92\)

- For \(95 - 99\): \(\frac { 95 + 99} { 2} = 97\)

- For \(100 - 104\): \(\frac { 100 + 104} { 2} = 102\)

- For \(105 - 109\): \(\frac { 105 + 109} { 2} = 107\)

- For \(110 - 114\): \(\frac { 110 + 114} { 2} = 112\)

Step 2: Multiply Midpoints by Frequencies

Next, multiply each midpoint by its corresponding frequency:

- \(72 \times 1 = 72\)

- \(77 \times 3 = 231\)

- \(82 \times 8 = 656\)

- \(87 \times 14 = 1218\)

- \(92 \times 19 = 1748\)

- \(97 \times 11 = 1067\)

- \(102 \times 9 = 918\)

- \(107 \times 5 = 535\)

- \(112 \times 1 = 112\)

Step 3: Sum the Products

Sum all the products from step 2:

\[72 + 231 + 656 + 1218 + 1748 + 1067 + 918 + 535 + 112 = 6557\]

Step 4: Divide by Total Frequency

Sum the frequencies:

\[1 + 3 + 8 + 14 + 19 + 11 + 9 + 5 + 1 = 71\]

Finally, divide the sum of the products by the total frequency:

\[\text { Mean} = \frac { 6557} { 71} \approx 92.49\]

So, the approximate mean for the given GFDT is \(\boxed{ 92.49} \).

Supplemental Knowledge

In statistics, a grouped frequency distribution table (GFDT) is used to organize large sets of data into classes or intervals. This method simplifies the data analysis process by grouping the data points into specified ranges and counting the number of occurrences within each range.

Key Concepts:

1. Class Interval: A range of values in which data is grouped.

2. Midpoint (Class Mark): The central value of a class interval, calculated as \(\frac { \text { Lower limit} + \text { Upper limit} } { 2} \).

3. Frequency: The number of data points within each class interval.

4. Mean Calculation in GFDT:

- Calculate midpoints for each class interval.

- Multiply each midpoint by its corresponding frequency to get the products.

- Sum all these products.

- Divide the sum by the total frequency.

Unleash Your Full Potential with UpStudy! Whether you need help solving complex statistical issues or homework in biology, UpStudy is here to provide instant, accurate solutions across multiple fields such as mathematics, chemistry, physics and biology - plus step-by-step explanations from elite tutors 24/7 for any difficult concepts! Join millions of other learners worldwide who rely on UpStudy as part of their educational success journey; download this app now to take your learning journey even further!