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Sagot :
To find the standard deviation of the given continuous series, follow these steps:
### Step 1: Identify the midpoints of each class interval
Each class interval is given, and we will first find the midpoints. A midpoint for a class interval [tex]\([a, b]\)[/tex] is calculated using the formula:
[tex]\[ \text{Midpoint} = \frac{a + b}{2} \][/tex]
So, for each class interval:
1. [tex]\(0-10\)[/tex]: Midpoint [tex]\(= \frac{0 + 10}{2} = 5\)[/tex]
2. [tex]\(10-20\)[/tex]: Midpoint [tex]\(= \frac{10 + 20}{2} = 15\)[/tex]
3. [tex]\(20-30\)[/tex]: Midpoint [tex]\(= \frac{20 + 30}{2} = 25\)[/tex]
4. [tex]\(30-40\)[/tex]: Midpoint [tex]\(= \frac{30 + 40}{2} = 35\)[/tex]
5. [tex]\(40-50\)[/tex]: Midpoint [tex]\(= \frac{40 + 50}{2} = 45\)[/tex]
Thus, the midpoints are [tex]\(5, 15, 25, 35, 45\)[/tex].
### Step 2: Calculate the mean of the series
The mean [tex]\(\overline{x}\)[/tex] is computed using the formula:
[tex]\[ \overline{x} = \frac{\sum (f \cdot m)}{\sum f} \][/tex]
where [tex]\(f\)[/tex] is the frequency and [tex]\(m\)[/tex] is the midpoint for each class interval.
Let's calculate [tex]\(\sum (f \cdot m)\)[/tex] and [tex]\(\sum f\)[/tex]:
[tex]\[ \begin{aligned} \sum (f \cdot m) &= (2 \cdot 5) + (6 \cdot 15) + (5 \cdot 25) + (4 \cdot 35) + (3 \cdot 45) \\ &= 10 + 90 + 125 + 140 + 135 \\ &= 500 \\ \sum f &= 2 + 6 + 5 + 4 + 3 = 20 \end{aligned} \][/tex]
Now, calculate the mean:
[tex]\[ \overline{x} = \frac{500}{20} = 25 \][/tex]
### Step 3: Calculate the variance
Variance [tex]\(\sigma^2\)[/tex] is calculated using the formula:
[tex]\[ \sigma^2 = \frac{\sum f (m - \overline{x})^2}{\sum f} \][/tex]
We need to find [tex]\((m - \overline{x})\)[/tex] and square those values, then multiply by their respective frequencies:
[tex]\[ \begin{aligned} \sum f (m - \overline{x})^2 &= 2(5 - 25)^2 + 6(15 - 25)^2 + 5(25 - 25)^2 + 4(35 - 25)^2 + 3(45 - 25)^2 \\ &= 2(-20)^2 + 6(-10)^2 + 5(0)^2 + 4(10)^2 + 3(20)^2 \\ &= 2(400) + 6(100) + 5(0) + 4(100) + 3(400) \\ &= 800 + 600 + 0 + 400 + 1200 \\ &= 3000 \end{aligned} \][/tex]
Now, calculate the variance:
[tex]\[ \sigma^2 = \frac{3000}{20} = 150 \][/tex]
### Step 4: Calculate the standard deviation
The standard deviation [tex]\(\sigma\)[/tex] is the square root of the variance:
[tex]\[ \sigma = \sqrt{150} \approx 12.247 \][/tex]
### Summary
- Midpoints: [tex]\(5.0, 15.0, 25.0, 35.0, 45.0\)[/tex]
- Mean: [tex]\(25.0\)[/tex]
- Variance: [tex]\(150.0\)[/tex]
- Standard deviation: [tex]\(12.247\)[/tex]
Thus, the standard deviation of the given continuous series is approximately [tex]\(12.247\)[/tex].
### Step 1: Identify the midpoints of each class interval
Each class interval is given, and we will first find the midpoints. A midpoint for a class interval [tex]\([a, b]\)[/tex] is calculated using the formula:
[tex]\[ \text{Midpoint} = \frac{a + b}{2} \][/tex]
So, for each class interval:
1. [tex]\(0-10\)[/tex]: Midpoint [tex]\(= \frac{0 + 10}{2} = 5\)[/tex]
2. [tex]\(10-20\)[/tex]: Midpoint [tex]\(= \frac{10 + 20}{2} = 15\)[/tex]
3. [tex]\(20-30\)[/tex]: Midpoint [tex]\(= \frac{20 + 30}{2} = 25\)[/tex]
4. [tex]\(30-40\)[/tex]: Midpoint [tex]\(= \frac{30 + 40}{2} = 35\)[/tex]
5. [tex]\(40-50\)[/tex]: Midpoint [tex]\(= \frac{40 + 50}{2} = 45\)[/tex]
Thus, the midpoints are [tex]\(5, 15, 25, 35, 45\)[/tex].
### Step 2: Calculate the mean of the series
The mean [tex]\(\overline{x}\)[/tex] is computed using the formula:
[tex]\[ \overline{x} = \frac{\sum (f \cdot m)}{\sum f} \][/tex]
where [tex]\(f\)[/tex] is the frequency and [tex]\(m\)[/tex] is the midpoint for each class interval.
Let's calculate [tex]\(\sum (f \cdot m)\)[/tex] and [tex]\(\sum f\)[/tex]:
[tex]\[ \begin{aligned} \sum (f \cdot m) &= (2 \cdot 5) + (6 \cdot 15) + (5 \cdot 25) + (4 \cdot 35) + (3 \cdot 45) \\ &= 10 + 90 + 125 + 140 + 135 \\ &= 500 \\ \sum f &= 2 + 6 + 5 + 4 + 3 = 20 \end{aligned} \][/tex]
Now, calculate the mean:
[tex]\[ \overline{x} = \frac{500}{20} = 25 \][/tex]
### Step 3: Calculate the variance
Variance [tex]\(\sigma^2\)[/tex] is calculated using the formula:
[tex]\[ \sigma^2 = \frac{\sum f (m - \overline{x})^2}{\sum f} \][/tex]
We need to find [tex]\((m - \overline{x})\)[/tex] and square those values, then multiply by their respective frequencies:
[tex]\[ \begin{aligned} \sum f (m - \overline{x})^2 &= 2(5 - 25)^2 + 6(15 - 25)^2 + 5(25 - 25)^2 + 4(35 - 25)^2 + 3(45 - 25)^2 \\ &= 2(-20)^2 + 6(-10)^2 + 5(0)^2 + 4(10)^2 + 3(20)^2 \\ &= 2(400) + 6(100) + 5(0) + 4(100) + 3(400) \\ &= 800 + 600 + 0 + 400 + 1200 \\ &= 3000 \end{aligned} \][/tex]
Now, calculate the variance:
[tex]\[ \sigma^2 = \frac{3000}{20} = 150 \][/tex]
### Step 4: Calculate the standard deviation
The standard deviation [tex]\(\sigma\)[/tex] is the square root of the variance:
[tex]\[ \sigma = \sqrt{150} \approx 12.247 \][/tex]
### Summary
- Midpoints: [tex]\(5.0, 15.0, 25.0, 35.0, 45.0\)[/tex]
- Mean: [tex]\(25.0\)[/tex]
- Variance: [tex]\(150.0\)[/tex]
- Standard deviation: [tex]\(12.247\)[/tex]
Thus, the standard deviation of the given continuous series is approximately [tex]\(12.247\)[/tex].
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