Question
The frequency table below shows the percentages obtained in a Biometry test for a random sample of 60 students:
Interval fi xi F
[30 – 40) 5 b1 c1
[40 – 50) 12 b2 c2
[50 – 60) 18 b3 c3
[60 – 70) 15 b4 c4
[70 – a1) 7 b5 c5
[a1 – 90) 2 b6 c6
[90 – 100) 1 b7 c7
For the percentages obtained, find the inter quartile range.
Interval fi xi F
[30 – 40) 5 b1 c1
[40 – 50) 12 b2 c2
[50 – 60) 18 b3 c3
[60 – 70) 15 b4 c4
[70 – a1) 7 b5 c5
[a1 – 90) 2 b6 c6
[90 – 100) 1 b7 c7
For the percentages obtained, find the inter quartile range.
Answers
GPT-4o mini
To find the interquartile range (IQR) from the given frequency table, we need to calculate the first quartile (Q1) and the third quartile (Q3) using the cumulative frequencies. The interquartile range is then calculated as \( IQR = Q3 - Q1 \).
First, we calculate the cumulative frequency \( F \) for each interval.
**Frequency Table with Cumulative Frequencies:**
- Interval [30 – 40):
\( f_1 = 5 \)
\( F_1 = 5 \)
- Interval [40 – 50):
\( f_2 = 12 \)
\( F_2 = F_1 + f_2 = 5 + 12 = 17 \)
- Interval [50 – 60):
\( f_3 = 18 \)
\( F_3 = F_2 + f_3 = 17 + 18 = 35 \)
- Interval [60 – 70):
\( f_4 = 15 \)
\( F_4 = F_3 + f_4 = 35 + 15 = 50 \)
- Interval [70 – 80):
\( f_5 = 7 \)
\( F_5 = F_4 + f_5 = 50 + 7 = 57 \)
- Interval [80 – 90):
\( f_6 = 2 \)
\( F_6 = F_5 + f_6 = 57 + 2 = 59 \)
- Interval [90 – 100):
\( f_7 = 1 \)
\( F_7 = F_6 + f_7 = 59 + 1 = 60 \)
Now, we summarize:
| Interval | \( f_i \) | \( F_i \) |
|-----------------|-----------|-----------|
| [30 – 40) | 5 | 5 |
| [40 – 50) | 12 | 17 |
| [50 – 60) | 18 | 35 |
| [60 – 70) | 15 | 50 |
| [70 – 80) | 7 | 57 |
| [80 – 90) | 2 | 59 |
| [90 – 100) | 1 | 60 |
### Calculate Q1 and Q3
1. **Finding Q1** (the 25th percentile):
\( \frac{N}{4} = \frac{60}{4} = 15 \)
We look for the cumulative frequency that is equal to or just exceeds 15:
- \( F_2 = 17 \): The 2nd interval \([40 - 50)\) is where Q1 lies.
To find the exact value of Q1:
\[
Q1 = L + \left(\frac{\frac{N}{4} - F}{f}\right) \cdot c
\]
Where:
- \( L = 40 \) (lower boundary of the interval where Q1 lies)
- \( F = 5 \) (cumulative frequency before this interval)
- \( f = 12 \) (frequency of the interval)
- \( c = 10 \) (class width - from 40 to 50)
Substituting into the formula:
\[
Q1 = 40 + \left(\frac{15 - 5}{12}\right) \cdot 10 = 40 + \left(\frac{10}{12}\right) \cdot 10 = 40 + \frac{100}{12} \approx 40 + 8.33 \approx 48.33
\]
2. **Finding Q3** (the 75th percentile):
\( \frac{3N}{4} = \frac{3 \cdot 60}{4} = 45 \)
We look for the cumulative frequency that is equal to or just exceeds 45:
- \( F_4 = 50 \): The 4th interval \([60 - 70)\) is where Q3 lies.
To find the exact value of Q3:
\[
Q3 = L + \left(\frac{\frac{3N}{4} - F}{f}\right) \cdot c
\]
Where:
- \( L = 60 \) (lower boundary of the interval where Q3 lies)
- \( F = 35 \) (cumulative frequency before this interval)
- \( f = 15 \) (frequency of the interval)
- \( c = 10 \) (class width - from 60 to 70)
Substituting into the formula:
\[
Q3 = 60 + \left(\frac{45 - 35}{15}\right) \cdot 10 = 60 + \left(\frac{10}{15}\right) \cdot 10 = 60 + \frac{100}{15} \approx 60 + 6.67 \approx 66.67
\]
### Calculate IQR
Finally, we calculate the interquartile range:
\[
IQR = Q3 - Q1 \approx 66.67 - 48.33 = 18.34
\]
The interquartile range (IQR) of the percentages obtained in the Biometry test is approximately **18.34**.
First, we calculate the cumulative frequency \( F \) for each interval.
**Frequency Table with Cumulative Frequencies:**
- Interval [30 – 40):
\( f_1 = 5 \)
\( F_1 = 5 \)
- Interval [40 – 50):
\( f_2 = 12 \)
\( F_2 = F_1 + f_2 = 5 + 12 = 17 \)
- Interval [50 – 60):
\( f_3 = 18 \)
\( F_3 = F_2 + f_3 = 17 + 18 = 35 \)
- Interval [60 – 70):
\( f_4 = 15 \)
\( F_4 = F_3 + f_4 = 35 + 15 = 50 \)
- Interval [70 – 80):
\( f_5 = 7 \)
\( F_5 = F_4 + f_5 = 50 + 7 = 57 \)
- Interval [80 – 90):
\( f_6 = 2 \)
\( F_6 = F_5 + f_6 = 57 + 2 = 59 \)
- Interval [90 – 100):
\( f_7 = 1 \)
\( F_7 = F_6 + f_7 = 59 + 1 = 60 \)
Now, we summarize:
| Interval | \( f_i \) | \( F_i \) |
|-----------------|-----------|-----------|
| [30 – 40) | 5 | 5 |
| [40 – 50) | 12 | 17 |
| [50 – 60) | 18 | 35 |
| [60 – 70) | 15 | 50 |
| [70 – 80) | 7 | 57 |
| [80 – 90) | 2 | 59 |
| [90 – 100) | 1 | 60 |
### Calculate Q1 and Q3
1. **Finding Q1** (the 25th percentile):
\( \frac{N}{4} = \frac{60}{4} = 15 \)
We look for the cumulative frequency that is equal to or just exceeds 15:
- \( F_2 = 17 \): The 2nd interval \([40 - 50)\) is where Q1 lies.
To find the exact value of Q1:
\[
Q1 = L + \left(\frac{\frac{N}{4} - F}{f}\right) \cdot c
\]
Where:
- \( L = 40 \) (lower boundary of the interval where Q1 lies)
- \( F = 5 \) (cumulative frequency before this interval)
- \( f = 12 \) (frequency of the interval)
- \( c = 10 \) (class width - from 40 to 50)
Substituting into the formula:
\[
Q1 = 40 + \left(\frac{15 - 5}{12}\right) \cdot 10 = 40 + \left(\frac{10}{12}\right) \cdot 10 = 40 + \frac{100}{12} \approx 40 + 8.33 \approx 48.33
\]
2. **Finding Q3** (the 75th percentile):
\( \frac{3N}{4} = \frac{3 \cdot 60}{4} = 45 \)
We look for the cumulative frequency that is equal to or just exceeds 45:
- \( F_4 = 50 \): The 4th interval \([60 - 70)\) is where Q3 lies.
To find the exact value of Q3:
\[
Q3 = L + \left(\frac{\frac{3N}{4} - F}{f}\right) \cdot c
\]
Where:
- \( L = 60 \) (lower boundary of the interval where Q3 lies)
- \( F = 35 \) (cumulative frequency before this interval)
- \( f = 15 \) (frequency of the interval)
- \( c = 10 \) (class width - from 60 to 70)
Substituting into the formula:
\[
Q3 = 60 + \left(\frac{45 - 35}{15}\right) \cdot 10 = 60 + \left(\frac{10}{15}\right) \cdot 10 = 60 + \frac{100}{15} \approx 60 + 6.67 \approx 66.67
\]
### Calculate IQR
Finally, we calculate the interquartile range:
\[
IQR = Q3 - Q1 \approx 66.67 - 48.33 = 18.34
\]
The interquartile range (IQR) of the percentages obtained in the Biometry test is approximately **18.34**.