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 sample standard deviation.

Answer 1

To find the sample standard deviation of the percentages obtained in the Biometry test given the frequency table, we will follow these steps:

1. **Calculate the midpoints (xi) of each interval**.
2. **Use the frequencies (fi) to calculate the mean**.
3. **Calculate the variance** using the mean and the frequencies.
4. **Find the sample standard deviation (SD)**.

### Step 1: Calculate midpoints (xi)

For each interval, we find the midpoint (xi) as follows:
- \([30, 40)\): \(x_1 = \frac{30 + 40}{2} = 35\)
- \([40, 50)\): \(x_2 = \frac{40 + 50}{2} = 45\)
- \([50, 60)\): \(x_3 = \frac{50 + 60}{2} = 55\)
- \([60, 70)\): \(x_4 = \frac{60 + 70}{2} = 65\)
- \([70, 80)\): \(x_5 = \frac{70 + 80}{2} = 75\)
- \([80, 90)\): \(x_6 = \frac{80 + 90}{2} = 85\)
- \([90, 100)\): \(x_7 = \frac{90 + 100}{2} = 95\)

### Step 2: Frequencies (fi) and Calculate Mean

The frequency \(fi\) for each interval is given:

| Interval | \(fi\) | \(xi\) |
| ----------- | ------ | ------ |
| [30, 40) | 5 | 35 |
| [40, 50) | 12 | 45 |
| [50, 60) | 18 | 55 |
| [60, 70) | 15 | 65 |
| [70, 80) | 7 | 75 |
| [80, 90) | 2 | 85 |
| [90, 100) | 1 | 95 |
| **Total** | 60 | |

To find the mean (\(\bar{x}\)), we first calculate the sum of \(fi \cdot xi\):

\[
\text{Sum} = (5 \cdot 35) + (12 \cdot 45) + (18 \cdot 55) + (15 \cdot 65) + (7 \cdot 75) + (2 \cdot 85) + (1 \cdot 95)
\]

Calculating each term:

- \(5 \cdot 35 = 175\)
- \(12 \cdot 45 = 540\)
- \(18 \cdot 55 = 990\)
- \(15 \cdot 65 = 975\)
- \(7 \cdot 75 = 525\)
- \(2 \cdot 85 = 170\)
- \(1 \cdot 95 = 95\)

\[
\text{Sum} = 175 + 540 + 990 + 975 + 525 + 170 + 95 = 3470
\]

Now, to calculate the mean:

\[
\bar{x} = \frac{\text{Sum}}{\text{Total}} = \frac{3470}{60} \approx 57.83
\]

### Step 3: Calculate Variance

We also need to calculate \(\sum (fi \cdot (xi - \bar{x})^2)\):

1. First find each \( (xi - \bar{x})^2 \):
- For \(x_1 = 35\): \((35 - 57.83)^2 \approx 522.52\)
- For \(x_2 = 45\): \((45 - 57.83)^2 \approx 164.72\)
- For \(x_3 = 55\): \((55 - 57.83)^2 \approx 8.16\)
- For \(x_4 = 65\): \((65 - 57.83)^2 \approx 72.67\)
- For \(x_5 = 75\): \((75 - 57.83)^2 \approx 293.34\)
- For \(x_6 = 85\): \((85 - 57.83)^2 \approx 723.31\)
- For \(x_7 = 95\): \((95 - 57.83)^2 \approx 1367.87\)

2. Now, calculate \(fi \cdot (xi - \bar{x})^2\):
- For \(f_1\): \(5 \cdot 522.52 \approx 2612.6\)
- For \(f_2\): \(12 \cdot 164.72 \approx 1976.64\)
- For \(f_3\): \(18 \cdot 8.16 \approx 146.88\)
- For \(f_4\): \(15 \cdot 72.67 \approx 1090.05\)
- For \(f_5\): \(7 \cdot 293.34 \approx 2053.38\)
- For \(f_6\): \(2 \cdot 723.31 \approx 1446.62\)
- For \(f_7\): \(1 \cdot 1367.87 \approx 1367.87\)

Finally, sum these values:

\[
\sum fi \cdot (xi - \bar{x})^2 \approx 2612.6 + 1976.64 + 146.88 + 1090.05 + 2053.38 + 1446.62 + 1367.87 \approx 10761.04
\]

Now calculate the variance (\(s^2\)):

\[
s^2 = \frac{\sum fi \cdot (xi - \bar{x})^2}{n-1} = \frac{10761.04}{60-1} \approx 183.14
\]

### Step 4: Find Sample Standard Deviation

Now find the sample standard deviation (\(s\)):

\[
s = \sqrt{s^2} = \sqrt{183.14} \approx 13.54
\]

Thus, the sample standard deviation for the percentages obtained is approximately **13.54**.