Question

the quantitative data was gathered by taking a random sample. calculate the standard deviation. round to one decimal place.
x
13
15
11
21
26

Explanation:

Step1: Calculate the mean ($\mu$)

The data set is \(13, 15, 11, 21, 26\). The number of data points \(n = 5\).
The mean \(\mu=\frac{13 + 15+11 + 21+26}{5}=\frac{86}{5} = 17.2\)

Step2: Calculate the squared differences from the mean

For \(x = 13\): \((13 - 17.2)^2=(- 4.2)^2 = 17.64\)
For \(x = 15\): \((15 - 17.2)^2=(-2.2)^2=4.84\)
For \(x = 11\): \((11 - 17.2)^2=(-6.2)^2 = 38.44\)
For \(x = 21\): \((21 - 17.2)^2=(3.8)^2 = 14.44\)
For \(x = 26\): \((26 - 17.2)^2=(8.8)^2=77.44\)

Step3: Calculate the variance ($\sigma^2$)

Variance is the average of the squared differences.
\(\sigma^2=\frac{17.64 + 4.84+38.44 + 14.44+77.44}{5}=\frac{152.8}{5}=30.56\)

Step4: Calculate the standard deviation ($\sigma$)

Standard deviation is the square root of the variance.
\(\sigma=\sqrt{30.56}\approx5.5\) (rounded to one decimal place)

Answer:

\(5.5\)