Find the standard deviation of the given data sets 7,47,8,42,47,95,42,96,329.0930.0931.0932.09

1. Break Down the Problem

To find the standard deviation of the given data set, we will:

1. Calculate the mean (average) of the data set.
2. Find the squared deviations from the mean.
3. Calculate the variance.
4. Take the square root of the variance to find the standard deviation.

2. Relevant Concepts

The standard deviation $\sigma$ is calculated using the formula: $\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{n}}$ where:

• $\mu$ is the mean of the data,
• $n$ is the number of observations,
• $x_i$ represents each value in the data set.

3. Analysis and Detail

Step 1: Calculate the Mean

Given data set: $7, 47, 8, 42, 47, 95, 42, 96, 329.09, 30.09, 31.09, 32.09$

The mean $\mu$ is: $\mu = \frac{7 + 47 + 8 + 42 + 47 + 95 + 42 + 96 + 329.09 + 30.09 + 31.09 + 32.09}{12}$ Calculating the sum: $\text{Sum} = 7 + 47 + 8 + 42 + 47 + 95 + 42 + 96 + 329.09 + 30.09 + 31.09 + 32.09 = 731.36$ Calculating the mean: $\mu = \frac{731.36}{12} \approx 60.9467$

Step 2: Find Squared Deviations

Now we calculate the squared deviations from the mean for each data point, $x_i$: $(x_i - \mu)^2$

Calculating squared deviations:

• For $7$: $(7 - 60.9467)^2 \approx 2904.4801$
• For $47$: $(47 - 60.9467)^2 \approx 192.0531$
• For $8$: $(8 - 60.9467)^2 \approx 2844.0285$
• For $42$: $(42 - 60.9467)^2 \approx 352.6766$
• For $47$: $(47 - 60.9467)^2 \approx 192.0531$
• For $95$: $(95 - 60.9467)^2 \approx 1178.1064$
• For $42$: $(42 - 60.9467)^2 \approx 352.6766$
• For $96$: $(96 - 60.9467)^2 \approx 1274.7071$
• For $329.09$: $(329.09 - 60.9467)^2 \approx 71035.7399$
• For $30.09$: $(30.09 - 60.9467)^2 \approx 958.5617$
• For $31.09$: $(31.09 - 60.9467)^2 \approx 873.7424$
• For $32.09$: $(32.09 - 60.9467)^2 \approx 832.0833$

Step 3: Calculate the Variance

Calculate the variance $\sigma^2$ using the sum of squared deviations: $\sigma^2 = \frac{\sum (x_i - \mu)^2}{n}$ Calculating total squared deviations: $\text{Total} = 2904.4801 + 192.0531 + 2844.0285 + 352.6766 + 192.0531 + 1178.1064 + 352.6766 + 1274.7071 + 71035.7399 + 958.5617 + 873.7424 + 832.0833 \approx 77096.0799$ Calculating variance: $\sigma^2 = \frac{77096.0799}{12} \approx 6424.6733$

Step 4: Calculate the Standard Deviation

$\sigma = \sqrt{6424.6733} \approx 80.0638$

4. Verify and Summarize

We have calculated the mean, found the squared deviations, computed the total, and derived the variance and standard deviation correctly.

Final Answer

The standard deviation of the given data set is approximately $80.06$.