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To find the standard deviation of the given frequency distribution, we need to follow several steps, including calculating the mean and variance as intermediate steps. Here is a detailed, step-by-step solution:
### Step 1: Calculate the Mean
First, we need to calculate the mean ([tex]\(\mu\)[/tex]) of the distribution.
The formula for the mean [tex]\(\mu\)[/tex] in a frequency distribution is:
[tex]\[ \mu = \frac{\sum (X_i \cdot f_i)}{\sum f_i} \][/tex]
where [tex]\(X_i\)[/tex] represents the values and [tex]\(f_i\)[/tex] represents the frequencies.
Given data:
- Values ([tex]\(X_i\)[/tex]): [12, 13, 14, 15, 16, 17]
- Frequencies ([tex]\(f_i\)[/tex]): [6, 6, 5, 1, 15, 12]
Calculate the products [tex]\(X_i \cdot f_i\)[/tex]:
[tex]\[ \begin{align*} 12 \cdot 6 &= 72 \\ 13 \cdot 6 &= 78 \\ 14 \cdot 5 &= 70 \\ 15 \cdot 1 &= 15 \\ 16 \cdot 15 &= 240 \\ 17 \cdot 12 &= 204 \\ \end{align*} \][/tex]
Sum these products:
[tex]\[ 72 + 78 + 70 + 15 + 240 + 204 = 679 \][/tex]
Sum of the frequencies:
[tex]\[ 6 + 6 + 5 + 1 + 15 + 12 = 45 \][/tex]
Now we can find the mean:
[tex]\[ \mu = \frac{679}{45} = 15.088888888888889 \][/tex]
### Step 2: Calculate the Variance
Next, we calculate the variance ([tex]\(\sigma^2\)[/tex]), the formula for variance in a frequency distribution is:
[tex]\[ \sigma^2 = \frac{\sum f_i (X_i - \mu)^2}{\sum f_i} \][/tex]
We need to find [tex]\((X_i - \mu)^2\)[/tex] and then multiply by [tex]\(f_i\)[/tex]:
[tex]\[ \begin{align*} (12 - 15.088888888888889)^2 &= 9.568641975308641 \\ (13 - 15.088888888888889)^2 &= 4.359753086419752 \\ (14 - 15.088888888888889)^2 &= 1.1854320987654307 \\ (15 - 15.088888888888889)^2 &= 0.007901234567901255 \\ (16 - 15.088888888888889)^2 &= 0.8320987654320987 \\ (17 - 15.088888888888889)^2 &= 3.6469135802469116 \\ \end{align*} \][/tex]
Multiply these squared differences by their respective frequencies and sum them:
[tex]\[ \begin{align*} 6 \cdot 9.568641975308641 &= 57.411851851851845 \\ 6 \cdot 4.359753086419752 &= 26.15851851851851 \\ 5 \cdot 1.1854320987654307 &= 5.927160493827153 \\ 1 \cdot 0.007901234567901255 &= 0.007901234567901255 \\ 15 \cdot 0.8320987654320987 &= 12.481481481481481 \\ 12 \cdot 3.6469135802469116 &= 43.762962962962946 \\ \end{align*} \][/tex]
Sum these products:
[tex]\[ 57.411851851851845 + 26.15851851851851 + 5.927160493827153 + 0.007901234567901255 + 12.481481481481481 + 43.762962962962946 = 145.74987654320987 \][/tex]
Now calculate the variance:
[tex]\[ \sigma^2 = \frac{145.74987654320987}{45} = 3.2365432098765434 \][/tex]
### Step 3: Calculate the Standard Deviation
Finally, the standard deviation ([tex]\(\sigma\)[/tex]) is the square root of the variance:
[tex]\[ \sigma = \sqrt{3.2365432098765434} = 1.7990395242674753 \][/tex]
So, the standard deviation of the given frequency distribution is approximately [tex]\(1.799\)[/tex].
### Step 1: Calculate the Mean
First, we need to calculate the mean ([tex]\(\mu\)[/tex]) of the distribution.
The formula for the mean [tex]\(\mu\)[/tex] in a frequency distribution is:
[tex]\[ \mu = \frac{\sum (X_i \cdot f_i)}{\sum f_i} \][/tex]
where [tex]\(X_i\)[/tex] represents the values and [tex]\(f_i\)[/tex] represents the frequencies.
Given data:
- Values ([tex]\(X_i\)[/tex]): [12, 13, 14, 15, 16, 17]
- Frequencies ([tex]\(f_i\)[/tex]): [6, 6, 5, 1, 15, 12]
Calculate the products [tex]\(X_i \cdot f_i\)[/tex]:
[tex]\[ \begin{align*} 12 \cdot 6 &= 72 \\ 13 \cdot 6 &= 78 \\ 14 \cdot 5 &= 70 \\ 15 \cdot 1 &= 15 \\ 16 \cdot 15 &= 240 \\ 17 \cdot 12 &= 204 \\ \end{align*} \][/tex]
Sum these products:
[tex]\[ 72 + 78 + 70 + 15 + 240 + 204 = 679 \][/tex]
Sum of the frequencies:
[tex]\[ 6 + 6 + 5 + 1 + 15 + 12 = 45 \][/tex]
Now we can find the mean:
[tex]\[ \mu = \frac{679}{45} = 15.088888888888889 \][/tex]
### Step 2: Calculate the Variance
Next, we calculate the variance ([tex]\(\sigma^2\)[/tex]), the formula for variance in a frequency distribution is:
[tex]\[ \sigma^2 = \frac{\sum f_i (X_i - \mu)^2}{\sum f_i} \][/tex]
We need to find [tex]\((X_i - \mu)^2\)[/tex] and then multiply by [tex]\(f_i\)[/tex]:
[tex]\[ \begin{align*} (12 - 15.088888888888889)^2 &= 9.568641975308641 \\ (13 - 15.088888888888889)^2 &= 4.359753086419752 \\ (14 - 15.088888888888889)^2 &= 1.1854320987654307 \\ (15 - 15.088888888888889)^2 &= 0.007901234567901255 \\ (16 - 15.088888888888889)^2 &= 0.8320987654320987 \\ (17 - 15.088888888888889)^2 &= 3.6469135802469116 \\ \end{align*} \][/tex]
Multiply these squared differences by their respective frequencies and sum them:
[tex]\[ \begin{align*} 6 \cdot 9.568641975308641 &= 57.411851851851845 \\ 6 \cdot 4.359753086419752 &= 26.15851851851851 \\ 5 \cdot 1.1854320987654307 &= 5.927160493827153 \\ 1 \cdot 0.007901234567901255 &= 0.007901234567901255 \\ 15 \cdot 0.8320987654320987 &= 12.481481481481481 \\ 12 \cdot 3.6469135802469116 &= 43.762962962962946 \\ \end{align*} \][/tex]
Sum these products:
[tex]\[ 57.411851851851845 + 26.15851851851851 + 5.927160493827153 + 0.007901234567901255 + 12.481481481481481 + 43.762962962962946 = 145.74987654320987 \][/tex]
Now calculate the variance:
[tex]\[ \sigma^2 = \frac{145.74987654320987}{45} = 3.2365432098765434 \][/tex]
### Step 3: Calculate the Standard Deviation
Finally, the standard deviation ([tex]\(\sigma\)[/tex]) is the square root of the variance:
[tex]\[ \sigma = \sqrt{3.2365432098765434} = 1.7990395242674753 \][/tex]
So, the standard deviation of the given frequency distribution is approximately [tex]\(1.799\)[/tex].
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