Question

11.4 statistics
score: 37.45/40 answered: 23/24
question 24
answer to two decimal places
find the mode:
find the range:
find the standard deviation:
question help: video
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Explanation:

Step1: Find the mode

The mode is the number that appears most frequently.
Given data: \(3,6,8,9,17,19,20,21,21,27\).
The number \(21\) appears twice, more frequently than other numbers.

Step2: Find the range

The range is calculated as \( \text{Range}=\text{Max}-\text{Min}\).
Here, \(\text{Max} = 27\), \(\text{Min}=3\).
So, \( \text{Range}=27 - 3=24\).

Step3: Find the standard deviation

The formula for the sample standard deviation \(s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}\).
First, calculate the mean \(\bar{x}=\frac{3+6+8+9+17+19+20+21+21+27}{10}=\frac{151}{10}=15.1\).
Then, \(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}=(3 - 15.1)^{2}+(6 - 15.1)^{2}+(8 - 15.1)^{2}+(9 - 15.1)^{2}+(17 - 15.1)^{2}+(19 - 15.1)^{2}+(20 - 15.1)^{2}+(21 - 15.1)^{2}+(21 - 15.1)^{2}+(27 - 15.1)^{2}\)
\(=(- 12.1)^{2}+(-9.1)^{2}+(-7.1)^{2}+(-6.1)^{2}+(1.9)^{2}+(3.9)^{2}+(4.9)^{2}+(5.9)^{2}+(5.9)^{2}+(11.9)^{2}\)
\(=146.41+82.81+50.41+37.21+3.61+15.21+24.01+34.81+34.81+141.61\)
\(=560.9\).
Since \(n = 10\), \(s=\sqrt{\frac{560.9}{9}}\approx\sqrt{62.3222}\approx7.91\).

Answer:

Mode: \(21\)
Range: \(24\)
Standard deviation: \(7.91\)