Question

for the following set of data, find the sample standard deviation, to the nearest thousandth.
data | frequency
0 | 6
3 | 5
5 | 2
8 | 1
9 | 2
12 | 4
13 | 2
14 | 6

Explanation:

Step1: Calculate total number of data points

First, we find the total number of data points \( n \) by summing the frequencies.
\( n = 6 + 5 + 2 + 1 + 2 + 4 + 2 + 6 = 28 \)

Step2: Calculate the mean \( \bar{x} \)

The formula for the mean with frequency is \( \bar{x} = \frac{\sum (x \cdot f)}{\sum f} \).
We calculate \( \sum (x \cdot f) \):

• For \( x = 0 \), \( 0 \cdot 6 = 0 \)
• For \( x = 3 \), \( 3 \cdot 5 = 15 \)
• For \( x = 5 \), \( 5 \cdot 2 = 10 \)
• For \( x = 8 \), \( 8 \cdot 1 = 8 \)
• For \( x = 9 \), \( 9 \cdot 2 = 18 \)
• For \( x = 12 \), \( 12 \cdot 4 = 48 \)
• For \( x = 13 \), \( 13 \cdot 2 = 26 \)
• For \( x = 14 \), \( 14 \cdot 6 = 84 \)

Summing these up: \( 0 + 15 + 10 + 8 + 18 + 48 + 26 + 84 = 219 \)

Then the mean \( \bar{x} = \frac{219}{28} \approx 7.8214 \)

Step3: Calculate the sum of squared deviations multiplied by frequency

The formula for sample standard deviation is \( s = \sqrt{\frac{\sum f(x - \bar{x})^2}{n - 1}} \).
We calculate \( (x - \bar{x})^2 \cdot f \) for each data point:

• For \( x = 0 \): \( (0 - 7.8214)^2 \cdot 6 \approx (-7.8214)^2 \cdot 6 \approx 61.174 \cdot 6 \approx 367.044 \)
• For \( x = 3 \): \( (3 - 7.8214)^2 \cdot 5 \approx (-4.8214)^2 \cdot 5 \approx 23.246 \cdot 5 \approx 116.23 \)
• For \( x = 5 \): \( (5 - 7.8214)^2 \cdot 2 \approx (-2.8214)^2 \cdot 2 \approx 7.960 \cdot 2 \approx 15.920 \)
• For \( x = 8 \): \( (8 - 7.8214)^2 \cdot 1 \approx (0.1786)^2 \cdot 1 \approx 0.0319 \cdot 1 \approx 0.0319 \)
• For \( x = 9 \): \( (9 - 7.8214)^2 \cdot 2 \approx (1.1786)^2 \cdot 2 \approx 1.389 \cdot 2 \approx 2.778 \)
• For \( x = 12 \): \( (12 - 7.8214)^2 \cdot 4 \approx (4.1786)^2 \cdot 4 \approx 17.461 \cdot 4 \approx 69.844 \)
• For \( x = 13 \): \( (13 - 7.8214)^2 \cdot 2 \approx (5.1786)^2 \cdot 2 \approx 26.818 \cdot 2 \approx 53.636 \)
• For \( x = 14 \): \( (14 - 7.8214)^2 \cdot 6 \approx (6.1786)^2 \cdot 6 \approx 38.175 \cdot 6 \approx 229.05 \)

Summing these up: \( 367.044 + 116.23 + 15.920 + 0.0319 + 2.778 + 69.844 + 53.636 + 229.05 \approx 854.5039 \)

Step4: Calculate the sample standard deviation

Now we use the formula \( s = \sqrt{\frac{\sum f(x - \bar{x})^2}{n - 1}} \).
We have \( \sum f(x - \bar{x})^2 \approx 854.5039 \) and \( n - 1 = 27 \)

So \( s = \sqrt{\frac{854.5039}{27}} \approx \sqrt{31.6483} \approx 5.6257 \approx 5.626 \) (to the nearest thousandth)

Answer:

\( 5.626 \)