Question

find (a) the range and (b) the standard deviation of the set of data. 23, 29, 21, 24, 23, 28, 27 (a) the range is 8. (b) the standard deviation is (round to the nearest thousandth as needed.)

Explanation:

Step1: Calculate the mean

The mean $\bar{x}=\frac{23 + 29+21 + 24+23+28+27}{7}=\frac{175}{7} = 25$

Step2: Calculate the squared differences from the mean

For \(23\): \((23 - 25)^2=(- 2)^2 = 4\)
For \(29\): \((29 - 25)^2=(4)^2 = 16\)
For \(21\): \((21 - 25)^2=(-4)^2 = 16\)
For \(24\): \((24 - 25)^2=(-1)^2 = 1\)
For \(23\): \((23 - 25)^2=(-2)^2 = 4\)
For \(28\): \((28 - 25)^2=(3)^2 = 9\)
For \(27\): \((27 - 25)^2=(2)^2 = 4\)

Step3: Calculate the variance

The variance \(s^{2}=\frac{4 + 16+16 + 1+4+9+4}{7}=\frac{54}{7}\approx7.714\)

Step4: Calculate the standard deviation

The standard deviation \(s=\sqrt{\frac{54}{7}}\approx2.777\)

Answer:

(a) The range is \(8\)
(b) The standard deviation is \(2.777\)