The following table of grouped data represents Mathematics score grade X
Score Frequency
40 - 44 2
45 - 49 8
50 - 54 15
55 - 59 10
60 - 64 5
65 - 69 10
a. Mean of the data is ...
b. Median of the data is ...
c. Mode of the data is ...
d. Q3 of the data is ... .

To find the **mean**, **median**, **mode**, and **third quartile (Q3)** of the grouped data, we will use the methods appropriate for grouped frequency distribution. Let's first organize the given data and then solve each part step by step.

| Score Range | Frequency (\(f\)) | Midpoint (\(x\)) | \(f \times x\) |
|-------------|--------------------|------------------|---------------|
| 40 - 44     | 2                  | 42               | 84            |
| 45 - 49     | 8                  | 47               | 376           |
| 50 - 54     | 15                 | 52               | 780           |
| 55 - 59     | 10                 | 57               | 570           |
| 60 - 64     | 5                  | 62               | 310           |
| 65 - 69     | 10                 | 67               | 670           |

Total frequency (\(N\)) = \(2 + 8 + 15 + 10 + 5 + 10 = 50\)

### (a) Mean of the data
The formula for the mean in grouped data is:
\[
\text{Mean} = \frac{\sum(f \times x)}{N}
\]
Where \(f\) is the frequency and \(x\) is the midpoint of each class.

Let's calculate the total of \(f \times x\):
\[
\sum(f \times x) = 84 + 376 + 780 + 570 + 310 + 670 = 2790
\]
Now, the mean is:
\[
\text{Mean} = \frac{2790}{50} = 55.8
\]

### (b) Median of the data
The median class is the class where the cumulative frequency reaches or exceeds \(\frac{N}{2} = \frac{50}{2} = 25\).

| Score Range | Frequency (\(f\)) | Cumulative Frequency |
|-------------|--------------------|----------------------|
| 40 - 44     | 2                  | 2                    |
| 45 - 49     | 8                  | 10                   |
| 50 - 54     | 15                 | 25                   |
| 55 - 59     | 10                 | 35                   |
| 60 - 64     | 5                  | 40                   |
| 65 - 69     | 10                 | 50                   |

The cumulative frequency exceeds 25 at the class 50 - 54, so this is the **median class**.

To calculate the median, we use the formula:
\[
\text{Median} = L + \left( \frac{\frac{N}{2} - F}{f_m} \right) \times h
\]
Where:
- \(L = 49.5\) (the lower boundary of the median class),
- \(N = 50\) (total frequency),
- \(F = 10\) (cumulative frequency before the median class),
- \(f_m = 15\) (frequency of the median class),
- \(h = 5\) (class width).

Substituting the values:
\[
\text{Median} = 49.5 + \left( \frac{25 - 10}{15} \right) \times 5 = 49.5 + \left( \frac{15}{15} \right) \times 5 = 49.5 + 5 = 54.5
\]

### (c) Mode of the data
The modal class is the class with the highest frequency. From the table, the highest frequency is \(15\) in the class 50 - 54, so this is the **modal class**.

To find the mode, we use the formula:
\[
\text{Mode} = L + \left( \frac{f_m - f_1}{(f_m - f_1) + (f_m - f_2)} \right) \times h
\]
Where:
- \(L = 49.5\) (lower boundary of the modal class),
- \(f_m = 15\) (frequency of the modal class),
- \(f_1 = 8\) (frequency of the class before the modal class),
- \(f_2 = 10\) (frequency of the class after the modal class),
- \(h = 5\) (class width).

Substituting the values:
\[
\text{Mode} = 49.5 + \left( \frac{15 - 8}{(15 - 8) + (15 - 10)} \right) \times 5 = 49.5 + \left( \frac{7}{7 + 5} \right) \times 5 = 49.5 + \left( \frac{7}{12} \right) \times 5
\]
\[
\text{Mode} = 49.5 + 2.92 = 52.42
\]

### (d) Third Quartile (Q3)
The third quartile (Q3) is the point where \(75\%\) of the data lies below. We calculate it using the formula:
\[
Q_3 = L + \left( \frac{\frac{3N}{4} - F}{f_Q} \right) \times h
\]
Where:
- \(L = 54.5\) (lower boundary of the class containing \(Q_3\)),
- \(N = 50\),
- \(\frac{3N}{4} = \frac{3 \times 50}{4} = 37.5\),
- \(F = 25\) (cumulative frequency before the class containing \(Q_3\)),
- \(f_Q = 10\) (frequency of the class containing \(Q_3\)),
- \(h = 5\) (class width).

Substituting the values:
\[
Q_3 = 54.5 + \left( \frac{37.5 - 25}{10} \right) \times 5 = 54.5 + \left( \frac{12.5}{10} \right) \times 5 = 54.5 + 6.25 = 60.75
\]

### Final Answers:
- **Mean**: 55.8
- **Median**: 54.5
- **Mode**: 52.42
- **Q3**: 60.75

Would you like more details or have any questions?

### Related Questions:
1. How do you calculate the mean if the frequencies are unequal in a dataset?
2. What is the significance of the median in a frequency distribution?
3. Can a dataset have more than one mode, and how do we handle such cases?
4. How is quartile used in interpreting data spread?
5. What adjustments are made for open-ended frequency classes?

#### Tip:
In grouped data, the more refined the grouping, the closer the estimated mean, median, and mode will be to the actual values.

Calculate the mean, median, mode, and third quartile (Q3) for a set of grouped data representing Grade X Mathematics scores. Understand how to apply the formulas for grouped frequency distributions, and learn step-by-step solutions for each measure of central tendency.

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