Question

1. 64, 69, 72, 54, 89, 92, 54, 32

32, 54, 54, 64, 69, 72, 90, 92
mean
65 - 75
median
66.5
mode
54
range
60
variance $\frac{sum(x - \bar{x})^2}{n}$
standard deviation $sqrt{\frac{sum(x - \bar{x})^2}{n}}$

Explanation:

Step1: Calculate the mean

The data set is \(32,54,54,64,69,72,89,92\). The mean \(\bar{x}=\frac{32 + 54+54+64+69+72+89+92}{8}=\frac{526}{8}=65.75\)

Step2: Calculate the median

Arrange the data in ascending - order: \(32,54,54,64,69,72,89,92\). Since \(n = 8\) (even), the median is the average of the \(\frac{n}{2}\)th and \((\frac{n}{2}+1)\)th values. \(\text{Median}=\frac{64 + 69}{2}=66.5\)

Step3: Find the mode

The mode is the number that appears most frequently. Here, \(54\) appears twice and other numbers appear once, so the mode is \(54\)

Step4: Calculate the range

The range is the difference between the largest and the smallest values. \(\text{Range}=92−32 = 60\)

Step5: Calculate the variance

First, find \((x-\bar{x})\) and \((x - \bar{x})^2\) for each \(x\) value:

\(x\)	\(\bar{x}\)	\(x-\bar{x}\)	\((x - \bar{x})^2\)
\(54\)	\(65.75\)	\(54 - 65.75=-11.75\)	\((-11.75)^2 = 138.0625\)
\(54\)	\(65.75\)	\(54 - 65.75=-11.75\)	\((-11.75)^2 = 138.0625\)
\(64\)	\(65.75\)	\(64 - 65.75=-1.75\)	\((-1.75)^2 = 3.0625\)
\(69\)	\(65.75\)	\(69 - 65.75 = 3.25\)	\((3.25)^2 = 10.5625\)
\(72\)	\(65.75\)	\(72 - 65.75 = 6.25\)	\((6.25)^2 = 39.0625\)
\(89\)	\(65.75\)	\(89 - 65.75 = 23.25\)	\((23.25)^2 = 540.5625\)
\(92\)	\(65.75\)	\(92 - 65.75 = 26.25\)	\((26.25)^2 = 689.0625\)

\(\sum(x-\bar{x})^2=1139.0625+138.0625 + 138.0625+3.0625+10.5625+39.0625+540.5625+689.0625 = 2797.5\)
\(\text{Variance}=\frac{\sum(x-\bar{x})^2}{n}=\frac{2797.5}{8}=349.6875\)

Step6: Calculate the standard deviation

\(\text{Standard Deviation}=\sqrt{\frac{\sum(x-\bar{x})^2}{n}}=\sqrt{349.6875}\approx18.699\)

Answer:

Mean: \(65.75\), Median: \(66.5\), Mode: \(54\), Range: \(60\), Variance: \(349.6875\), Standard Deviation: approximately \(18.70\)