
Grouped Frequency Distribution Table
Scores
(Apparent limits)

Real limits

Midpoint

f

Relative f

cf

Percentile

85 - 91

84.5 - 91.5

88

20

0.2174

92

100%

78 - 84

77.5 - 84.5

81

2

0.0217

72

78.2609%

71 - 77

70.5 - 77.5

74

2

0.0217

70

76.087%

64 - 70

63.5 - 70.5

67

21

0.2283

68

  
Number
 %

57 - 63

56.5 - 63.5

60

2

0.0217

47

51.087%

50 - 56

49.5 - 56.5

53

16

0.1739

45

48.913%

43 - 49

42.5 - 49.5

46

2

0.0217

29

31.5217%

36 - 42

35.5 - 42.5

39

5

0.0543

27

29.3478%

29 - 35

28.5 - 35.5

32

12

0.1304

22

23.913%

22 - 28

21.5 - 28.5

25

10

0.1087

10

10.8696%

Using the midpoint value, identify the median score:    
Number
 

Using the midpoint value, identify Q1:   
Number
 

Using the midpoint value, identify Q3:   
Number


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.

Using the midpoint value, identify the median score, Q1, and Q3 from the grouped frequency distribution table provided by the user.

This page provides a step-by-step explanation of how to calculate the median, Q1 (25th percentile), and Q3 (75th percentile) from a grouped frequency distribution table. Learn to use midpoint values, cumulative frequencies, and interpolation methods. Suitable for high school students in Grades 10-12.

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