To solve the problem, we need to calculate the
mean,
variance, and
standard deviation of a given dataset. However, since the image does not provide the actual dataset (the values), I will explain the general steps to solve this problem using a hypothetical dataset. Let's assume the dataset is:
$$
\{5, 8, 10, 12, 15\}
$$
Step 1: Calculate the Mean ($ \bar{x} $)
The mean is calculated as the sum of all values divided by the number of values ($ n $).
$$
\bar{x} = \frac{\sum x_i}{n}
$$
For our dataset:
$$
\bar{x} = \frac{5 + 8 + 10 + 12 + 15}{5} = \frac{50}{5} = 10
$$
So, the mean is:
$$
\bar{x} = 10
$$
Step 2: Calculate the Differences from the Mean ($ x_i - \bar{x} $)
For each value $ x_i $ in the dataset, subtract the mean ($ \bar{x} $):
- For $ x_1 = 5 $: $ 5 - 10 = -5 $
- For $ x_2 = 8 $: $ 8 - 10 = -2 $
- For $ x_3 = 10 $: $ 10 - 10 = 0 $
- For $ x_4 = 12 $: $ 12 - 10 = 2 $
- For $ x_5 = 15 $: $ 15 - 10 = 5 $
The differences are:
$$
-5, -2, 0, 2, 5
$$
Step 3: Calculate the Squared Differences from the Mean ($ (x_i - \bar{x})^2 $)
Square each of the differences:
- For $ -5 $: $ (-5)^2 = 25 $
- For $ -2 $: $ (-2)^2 = 4 $
- For $ 0 $: $ 0^2 = 0 $
- For $ 2 $: $ 2^2 = 4 $
- For $ 5 $: $ 5^2 = 25 $
The squared differences are:
$$
25, 4, 0, 4, 25
$$
Step 4: Calculate the Sum of the Squared Differences
Sum the squared differences:
$$
\sum (x_i - \bar{x})^2 = 25 + 4 + 0 + 4 + 25 = 58
$$
Step 5: Calculate the Variance
The variance is the sum of the squared differences divided by the degrees of freedom ($ n - 1 $), where $ n $ is the number of values.
$$
\text{Variance} = \frac{\sum (x_i - \bar{x})^2}{n - 1}
$$
For our dataset:
$$
\text{Variance} = \frac{58}{5 - 1} = \frac{58}{4} = 14.5
$$
Step 6: Calculate the Standard Deviation
The standard deviation is the square root of the variance.
$$
\text{Standard Deviation} = \sqrt{\text{Variance}}
$$
For our dataset:
$$
\text{Standard Deviation} = \sqrt{14.5} \approx 3.81
$$
Final Answer
Putting it all together:
- Mean ($ \bar{x} $): $ 10 $
- Variance: $ 14.5 $
- Standard Deviation: $ \sqrt{14.5} \approx 3.81 $
$$
\boxed{10, 14.5, 3.81}
$$
If you provide the actual dataset, I can perform the calculations for that specific data.
Parent Tip: Review the logic above to help your child master the concept of standard deviation worksheet.
