Question

Calculate the standard deviation of the data set. Round to the nearest tenth. {20, 35, 41, 48, 79, 83, 83, 97, 120}
Responses 79.0 79.0 32.9 32.9 29.8 29.8 31.0

Answer 1

To calculate the standard deviation of the data set {20, 35, 41, 48, 79, 83, 83, 97, 120}, we will follow these steps:

1. **Calculate the mean (average)** of the data set.
2. **Subtract the mean** from each data point and square the result (this is the squared deviation).
3. **Calculate the average of the squared deviations**.
4. **Take the square root** of the average from step 3 to get the standard deviation.

### Step 1: Calculate the Mean
\[
\text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}} = \frac{20 + 35 + 41 + 48 + 79 + 83 + 83 + 97 + 120}{9}
\]
\[
= \frac{506}{9} \approx 56.222
\]

### Step 2: Calculate the Squared Deviations
Now we will subtract the mean from each data point and square each result.

- \( (20 - 56.222)^2 \approx 1295.656 \)
- \( (35 - 56.222)^2 \approx 448.163 \)
- \( (41 - 56.222)^2 \approx 231.856 \)
- \( (48 - 56.222)^2 \approx 66.665 \)
- \( (79 - 56.222)^2 \approx 517.215 \)
- \( (83 - 56.222)^2 \approx 719.303 \)
- \( (83 - 56.222)^2 \approx 719.303 \) (repeated since 83 appears twice)
- \( (97 - 56.222)^2 \approx 1679.039 \)
- \( (120 - 56.222)^2 \approx 3986.456 \)

### Step 3: Calculate the Variance
Now we will sum the squared deviations and divide by the number of data points (since this is a population version of standard deviation).

\[
\text{Variance} = \frac{(1295.656 + 448.163 + 231.856 + 66.665 + 517.215 + 719.303 + 719.303 + 1679.039 + 3986.456)}{9}
\]

Calculating the total:
\[
= 1295.656 + 448.163 + 231.856 + 66.665 + 517.215 + 719.303 + 719.303 + 1679.039 + 3986.456 = 10310.655
\]

Now dividing by 9:
\[
\text{Variance} \approx \frac{10310.655}{9} \approx 1145.628
\]

### Step 4: Calculate the Standard Deviation
Finally, we take the square root of the variance to get the standard deviation:
\[
\text{Standard Deviation} = \sqrt{1145.628} \approx 33.9
\]

Rounding to the nearest tenth, the standard deviation of the data set is approximately:

\[
\boxed{33.9}
\]