To solve the given problems step by step, we will first calculate the variance and standard deviation for the first set of data, and then we will do the same for the second set of data.
### Part (a): Finding Variance and Standard Deviation
**Step 1: Write down the data.**
The data given is:
65, 58, 68, 44, 48, 45, 60, 62, 60, 50
**Step 2: Calculate the mean (x̄).**
To find the mean, we sum all the observations and divide by the number of observations.
\[
\text{Mean} (x̄) = \frac{65 + 58 + 68 + 44 + 48 + 45 + 60 + 62 + 60 + 50}{10} = \frac{560}{10} = 56
\]
**Step 3: Calculate \(x_i - x̄\) for each observation.**
Now, subtract the mean from each observation:
- \(65 - 56 = 9\)
- \(58 - 56 = 2\)
- \(68 - 56 = 12\)
- \(44 - 56 = -12\)
- \(48 - 56 = -8\)
- \(45 - 56 = -11\)
- \(60 - 56 = 4\)
- \(62 - 56 = 6\)
- \(60 - 56 = 4\)
- \(50 - 56 = -6\)
**Step 4: Calculate \((x_i - x̄)^2\) for each observation.**
Now, square each of the differences:
- \(9^2 = 81\)
- \(2^2 = 4\)
- \(12^2 = 144\)
- \((-12)^2 = 144\)
- \((-8)^2 = 64\)
- \((-11)^2 = 121\)
- \(4^2 = 16\)
- \(6^2 = 36\)
- \(4^2 = 16\)
- \((-6)^2 = 36\)
**Step 5: Sum the squared differences.**
Now, sum all the squared differences:
\[
81 + 4 + 144 + 144 + 64 + 121 + 16 + 36 + 16 + 36 = 536
\]
**Step 6: Calculate the variance.**
Variance (\(σ^2\)) is calculated by dividing the sum of squared differences by the number of observations (n):
\[
\text{Variance} (σ^2) = \frac{536}{10} = 53.6
\]
**Step 7: Calculate the standard deviation.**
Standard deviation (\(σ\)) is the square root of the variance:
\[
\text{Standard Deviation} (σ) = \sqrt{53.6} \approx 7.33
\]
### Summary for Part (a):
- Variance = 53.6
- Standard Deviation ≈ 7.33
---
### Part (b): Finding Variance for Batsman A
**Step 1: Write down the data.**
The scores of batsman A are:
48, 80, 58, 44, 52, 65, 73, 56, 64, 50
**Step 2: Calculate the mean (x̄).**
To find the mean, we sum all the observations and divide by the number of observations.
\[
\text{Mean} (x̄) = \frac{48 + 80 + 58 + 44 + 52 + 65 + 73 + 56 + 64 + 50}{10} = \frac{590}{10} = 59
\]
**Step 3: Calculate \(x_i - x̄\) for each observation.**
Now, subtract the mean from each observation:
- \(48 - 59 = -11\)
- \(80 - 59 = 21\)
- \(58 - 59 = -1\)
- \(44 - 59 = -15\)
- \(52 - 59 = -7\)
- \(65 - 59 = 6\)
- \(73 - 59 = 14\)
- \(56 - 59 = -3\)
- \(64 - 59 = 5\)
- \(50 - 59 = -9\)
**Step 4: Calculate \((x_i - x̄)^2\) for each observation.**
Now, square each of the differences:
- \((-11)^2 = 121\)
- \(21^2 = 441\)
- \((-1)^2 = 1\)
- \((-15)^2 = 225\)
- \((-7)^2 = 49\)
- \(6^2 = 36\)
- \(14^2 = 196\)
- \((-3)^2 = 9\)
- \(5^2 = 25\)
- \((-9)^2 = 81\)
**Step 5: Sum the squared differences.**
Now, sum all the squared differences:
\[
121 + 441 + 1 + 225 + 49 + 36 + 196 + 9 + 25 + 81 = 1184
\]
**Step 6: Calculate the variance.**
Variance (\(σ^2\)) is calculated by dividing the sum of squared differences by the number of observations (n):
\[
\text{Variance} (σ^2) = \frac{1184}{10} = 118.4
\]
### Summary for Part (b):
- Variance = 118.4
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