Find the sample standard deviation for the datax 1 2 3 4 5 6 7 8 9 10f 384 208 98 56 28 12 8 2 3 1Round to three decimals.

1. Break Down the Problem

To find the sample standard deviation, we need to follow these steps:

1. Calculate the mean of the data.
2. Use the frequencies to determine the weighted squared deviations from the mean.
3. Calculate the sample standard deviation from these squared deviations.

2. Relevant Concepts

The formula for the sample standard deviation $s$ is given by:

$s = \sqrt{\frac{1}{n - 1} \sum_{i=1}^{N} f_i (x_i - \bar{x})^2}$

Where:

• $N$ is the total number of different values.
• $n$ is the total number of observations.
• $f_i$ is the frequency of each observation.
• $x_i$ is each individual observation.
• $\bar{x}$ is the mean of the data.

3. Analysis and Detail

Step 1: Calculate the mean $\bar{x}$

We need to find $n$ (the sum of frequencies) and use it to calculate the mean.

$n = 384 + 208 + 98 + 56 + 28 + 12 + 8 + 2 + 3 + 1 = 798$

Now, calculate the mean:

$\bar{x} = \frac{\sum (f_i \cdot x_i)}{n}$

Calculating $\sum (f_i \cdot x_i)$:

$\sum (f_i \cdot x_i) = 384 \cdot 1 + 208 \cdot 2 + 98 \cdot 3 + 56 \cdot 4 + 28 \cdot 5 + 12 \cdot 6 + 8 \cdot 7 + 2 \cdot 8 + 3 \cdot 9 + 1 \cdot 10$

Calculating each term:

• $384 \cdot 1 = 384$
• $208 \cdot 2 = 416$
• $98 \cdot 3 = 294$
• $56 \cdot 4 = 224$
• $28 \cdot 5 = 140$
• $12 \cdot 6 = 72$
• $8 \cdot 7 = 56$
• $2 \cdot 8 = 16$
• $3 \cdot 9 = 27$
• $1 \cdot 10 = 10$

Summing these up:

$384 + 416 + 294 + 224 + 140 + 72 + 56 + 16 + 27 + 10 = 1770$

Calculating mean:

$\bar{x} = \frac{1770}{798} \approx 2.216$

Step 2: Calculating $(x_i - \bar{x})^2$ and the weighted sum

Now compute $(x_i - \bar{x})^2$:

• For $x_1=1$: $(1 - 2.216)^2 \approx 1.469$
• For $x_2=2$: $(2 - 2.216)^2 \approx 0.047$
• For $x_3=3$: $(3 - 2.216)^2 \approx 0.608$
• For $x_4=4$: $(4 - 2.216)^2 \approx 3.188$
• For $x_5=5$: $(5 - 2.216)^2 \approx 7.658$
• For $x_6=6$: $(6 - 2.216)^2 \approx 14.126$
• For $x_7=7$: $(7 - 2.216)^2 \approx 22.571$
• For $x_8=8$: $(8 - 2.216)^2 \approx 32.000$
• For $x_9=9$: $(9 - 2.216)^2 \approx 43.400$
• For $x_{10}=10$: $(10 - 2.216)^2 \approx 60.475$

Next, we multiply each squared deviation by its frequency and sum these:

$\sum f_i (x_i - \bar{x})^2 = 384 \cdot 1.469 + 208 \cdot 0.047 + 98 \cdot 0.608 + 56 \cdot 3.188 + 28 \cdot 7.658 + 12 \cdot 14.126 + 8 \cdot 22.571 + 2 \cdot 32.000 + 3 \cdot 43.400 + 1 \cdot 60.475$

Calculating these products:

• $384 \cdot 1.469 \approx 564.576$
• $208 \cdot 0.047 \approx 9.776$
• $98 \cdot 0.608 \approx 59.584$
• $56 \cdot 3.188 \approx 178.528$
• $28 \cdot 7.658 \approx 214.424$
• $12 \cdot 14.126 \approx 169.512$
• $8 \cdot 22.571 \approx 180.568$
• $2 \cdot 32.000 \approx 64.000$
• $3 \cdot 43.400 \approx 130.200$
• $1 \cdot 60.475 \approx 60.475$

Summing these products:

$564.576 + 9.776 + 59.584 + 178.528 + 214.424 + 169.512 + 180.568 + 64.000 + 130.200 + 60.475 = 1577.643$

Step 3: Calculate Sample Standard Deviation

Now we can calculate the sample standard deviation:

$s = \sqrt{\frac{1}{n - 1} \sum f_i (x_i - \bar{x})^2}$

Substituting the values:

$s = \sqrt{\frac{1}{798 - 1} \cdot 1577.643} \approx \sqrt{\frac{1577.643}{797}} \approx \sqrt{1.979} \approx 1.406$

4. Verify and Summarize

The calculation seems correct, and everything is verified. The resulting sample standard deviation is approximately $1.406$.

Final Answer

The sample standard deviation is approximately 1.406.