To find the median, Q1 (the 25th percentile), and Q3 (the 75th percentile) using the midpoint values, we will need to calculate their corresponding cumulative frequencies (cf) and percentiles from the provided grouped frequency distribution table.

1. Finding the Median (50th percentile):

The cumulative frequency nearest to 50% corresponds to the median score.

• From the table, at a cumulative frequency of 45, the percentile is 48.913% (which corresponds to the class interval 50-56, with a midpoint of 53).
• At the next cumulative frequency of 47, the percentile is 51.087% (class interval 57-63, with a midpoint of 60).

Since 50% lies between these two cumulative frequencies (45 and 47), the median score lies between 53 and 60.

We can interpolate between these two values to estimate the median. Since 50% is closer to 51.087% (cf = 47), the median score is approximately 60.

2. Finding Q1 (25th percentile):

We look for the cumulative frequency nearest to 25%.

• At a cumulative frequency of 22, the percentile is 23.913% (class interval 29-35, midpoint 32).
• At the next cumulative frequency of 27, the percentile is 29.3478% (class interval 36-42, midpoint 39).

Since the 25th percentile lies between these two cumulative frequencies (22 and 27), we interpolate to estimate Q1. Since 25% is closer to 23.913%, the Q1 value is approximately 32.

3. Finding Q3 (75th percentile):

We look for the cumulative frequency nearest to 75%.

• At a cumulative frequency of 68, the percentile is not given in the table, but we know it is greater than 51.087%.
• By estimating from the trend in cumulative frequencies, the 75th percentile lies close to a cumulative frequency around 70, which corresponds to the class interval 71-77, with a midpoint of 74.

Thus, the Q3 value is approximately 74.

Summary:

• Median score (50th percentile): ≈ 60
• Q1 (25th percentile): ≈ 32
• Q3 (75th percentile): ≈ 74

Would you like further details or clarification on any step?

5 Related Questions:

1. How is the cumulative frequency (cf) calculated for grouped data?
2. Why do we use interpolation to find the median in grouped data?
3. How are real limits different from apparent limits in a frequency table?
4. Can we calculate the exact quartiles without interpolation?
5. How is the relative frequency (f) used in interpreting data distributions?

Tip:

When calculating percentiles in grouped data, interpolation between class intervals gives more accurate estimates than using just midpoint values.