To determine the standard deviation of the grouped sample data, follow these steps:
### Step 1: Calculate the midpoints of each interval.
The midpoint of an interval [tex]\([a, b]\)[/tex] is given by:
[tex]\[ \text{Midpoint} = \frac{a + b}{2} \][/tex]
Given the intervals:
- [tex]\((0.5, 3.5)\)[/tex]: Midpoint = [tex]\(\frac{0.5 + 3.5}{2} = 2.0\)[/tex]
- [tex]\((3.5, 6.5)\)[/tex]: Midpoint = [tex]\(\frac{3.5 + 6.5}{2} = 5.0\)[/tex]
- [tex]\((6.5, 9.5)\)[/tex]: Midpoint = [tex]\(\frac{6.5 + 9.5}{2} = 8.0\)[/tex]
- [tex]\((9.5, 12.5)\)[/tex]: Midpoint = [tex]\(\frac{9.5 + 12.5}{2} = 11.0\)[/tex]
So, the midpoints are: [tex]\([2.0, 5.0, 8.0, 11.0]\)[/tex].
### Step 2: Calculate the mean ([tex]\(\bar{x}\)[/tex]) of the grouped data.
The mean is calculated using the formula:
[tex]\[ \bar{x} = \frac{\sum (f \cdot x)}{\sum f} \][/tex]
Where [tex]\(f\)[/tex] is the frequency, and [tex]\(x\)[/tex] is the midpoint.
Using the data:
- Frequencies: [tex]\([3, 5, 4, 5]\)[/tex]
- Midpoints: [tex]\([2.0, 5.0, 8.0, 11.0]\)[/tex]
The weighted sums are:
- [tex]\(3 \cdot 2.0 = 6.0\)[/tex]
- [tex]\(5 \cdot 5.0 = 25.0\)[/tex]
- [tex]\(4 \cdot 8.0 = 32.0\)[/tex]
- [tex]\(5 \cdot 11.0 = 55.0\)[/tex]
Sum of all midpoints times their frequencies:
[tex]\[ 6.0 + 25.0 + 32.0 + 55.0 = 118.0 \][/tex]
Sum of frequencies:
[tex]\[ 3 + 5 + 4 + 5 = 17 \][/tex]
Therefore, the mean is:
[tex]\[ \bar{x} = \frac{118.0}{17} \approx 6.94 \][/tex]
### Step 3: Calculate the variance ([tex]\(\sigma^2\)[/tex]) of the grouped data.
The variance is calculated using the formula:
[tex]\[ \sigma^2 = \frac{\sum (f \cdot (x - \bar{x})^2)}{\sum f} \][/tex]
First, find each squared difference from the mean, weighted by the frequency:
- For midpoint 2.0:
[tex]\[ f \cdot (x - \bar{x})^2 = 3 \cdot (2.0 - 6.94)^2 = 3 \cdot (-4.94)^2 = 3 \cdot 24.40 = 73.21 \][/tex]
- For midpoint 5.0:
[tex]\[ f \cdot (x - \bar{x})^2 = 5 \cdot (5.0 - 6.94)^2 = 5 \cdot (-1.94)^2 = 5 \cdot 3.76 = 18.84 \][/tex]
- For midpoint 8.0:
[tex]\[ f \cdot (x - \bar{x})^2 = 4 \cdot (8.0 - 6.94)^2 = 4 \cdot (1.06)^2 = 4 \cdot 1.12 = 4.47 \][/tex]
- For midpoint 11.0:
[tex]\[ f \cdot (x - \bar{x})^2 = 5 \cdot (11.0 - 6.94)^2 = 5 \cdot (4.06)^2 = 5 \cdot 16.48 = 82.98 \][/tex]
Sum of all weighted squared differences:
[tex]\[ 73.21 + 18.84 + 4.47 + 82.98 = 179.50 \][/tex]
Variance:
[tex]\[ \sigma^2 = \frac{179.50}{17} \approx 10.53 \][/tex]
### Step 4: Calculate the standard deviation ([tex]\(\sigma\)[/tex]).
The standard deviation is the square root of the variance:
[tex]\[ \sigma = \sqrt{10.53} \approx 3.24 \][/tex]
Therefore, the standard deviation of the grouped sample data is:
[tex]\[ s \approx 3.24 \][/tex]
