Alright, let's work through this problem step-by-step to find the standard deviation for the given set of grouped data.
First, we need to gather the given data:
| Interval | Frequency (f) |
|-----------|---------------|
| 0.5-3.5 | 6 |
| 3.5-6.5 | 5 |
| 6.5-9.5 | 4 |
| 9.5-12.5 | 5 |
1. Find the midpoints of each interval:
For each interval, we calculate the midpoint by averaging the lower and upper bounds of the interval.
[tex]\[
\text{Midpoints:}
\][/tex]
[tex]\[
\begin{align*}
(0.5 + 3.5) / 2 &= 2.0 \\
(3.5 + 6.5) / 2 &= 5.0 \\
(6.5 + 9.5) / 2 &= 8.0 \\
(9.5 + 12.5) / 2 &= 11.0 \\
\end{align*}
\][/tex]
2. Calculate the weighted mean:
To find the mean, we use the sum of the products of each midpoint and its corresponding frequency, divided by the total number of data points.
Total frequency (total number of data points) [tex]\( N \)[/tex]:
[tex]\[
N = 6 + 5 + 4 + 5 = 20
\][/tex]
Weighted mean formula:
[tex]\[
\bar{x} = \frac{\sum (f \cdot x)}{N}
\][/tex]
Where [tex]\( x \)[/tex] is the midpoint, and [tex]\( f \)[/tex] is the frequency.
[tex]\[
\bar{x} = \frac{(6 \cdot 2.0) + (5 \cdot 5.0) + (4 \cdot 8.0) + (5 \cdot 11.0)}{20}
\][/tex]
[tex]\[
\bar{x} = \frac{12 + 25 + 32 + 55}{20} = \frac{124}{20} = 6.2
\][/tex]
3. Calculate the variance:
Variance formula for grouped data:
[tex]\[
\sigma^2 = \frac{\sum f \cdot (x - \bar{x})^2}{N}
\][/tex]
Where [tex]\( x \)[/tex] is the midpoint, [tex]\( \bar{x} \)[/tex] is the mean, and [tex]\( f \)[/tex] is the frequency.
[tex]\[
\begin{align*}
\sigma^2 &= \frac{6 \cdot (2.0 - 6.2)^2 + 5 \cdot (5.0 - 6.2)^2 + 4 \cdot (8.0 - 6.2)^2 + 5 \cdot (11.0 - 6.2)^2}{20}
\end{align*}
\][/tex]
Calculating each term inside the sum:
[tex]\[
\begin{align*}
6 \cdot (2.0 - 6.2)^2 &= 6 \cdot (-4.2)^2 = 6 \cdot 17.64 = 105.84 \\
5 \cdot (5.0 - 6.2)^2 &= 5 \cdot (-1.2)^2 = 5 \cdot 1.44 = 7.2 \\
4 \cdot (8.0 - 6.2)^2 &= 4 \cdot 1.8^2 = 4 \cdot 3.24 = 12.96 \\
5 \cdot (11.0 - 6.2)^2 &= 5 \cdot 4.8^2 = 5 \cdot 23.04 = 115.2 \\
\end{align*}
\][/tex]
Adding these together:
[tex]\[
105.84 + 7.2 + 12.96 + 115.2 = 241.2
\][/tex]
Dividing by the total number of data points [tex]\( N \)[/tex]:
[tex]\[
\sigma^2 = \frac{241.2}{20} = 12.06
\][/tex]
4. Calculate the standard deviation:
Standard deviation [tex]\( \sigma \)[/tex] is the square root of the variance:
[tex]\[
\sigma = \sqrt{12.06} = 3.47
\][/tex]
So, the standard deviation for the given set of grouped sample data, rounded to two decimal places, is [tex]\( \boxed{3.47} \)[/tex].
