Question

76
48
67
89
64
67
93
48
77
81
statistics
measures of center
mean=
median =
mode =
measures of spread
range =
standard deviation (round to 1 decimal place) =
interquartile range, iqr =
5 - number summary
minimum =
lower quartile, q1 =
median =
upper quartile, q3 =
maximum =

Explanation:

Step1: Sort the data

First, sort the data set \(48,48,64,67,67,76,77,81,89,93\)

Step2: Calculate the mean

The mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\), where \(n = 10\) and \(\sum_{i=1}^{10}x_{i}=48 + 48+64+67+67+76+77+81+89+93=710\)
\(\bar{x}=\frac{710}{10}=71\)

Step3: Calculate the median

Since \(n = 10\) (even), the median \(M=\frac{x_{\frac{n}{2}}+x_{\frac{n}{2}+1}}{2}\)
\(x_{5}=67\), \(x_{6}=76\)
\(M=\frac{67 + 76}{2}=71.5\)

Step4: Find the mode

The mode is the most - frequently occurring value. Here, \(48\) and \(67\) both occur twice.

Step5: Calculate the range

The range \(R=\text{Max}-\text{Min}\)
\(\text{Max}=93\), \(\text{Min}=48\)
\(R=93 - 48=45\)

Step6: Calculate the standard deviation

The formula for the sample standard deviation \(s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}\)
\(\sum_{i=1}^{10}(x_{i}-71)^{2}=(48 - 71)^{2}+(48 - 71)^{2}+(64 - 71)^{2}+(67 - 71)^{2}+(67 - 71)^{2}+(76 - 71)^{2}+(77 - 71)^{2}+(81 - 71)^{2}+(89 - 71)^{2}+(93 - 71)^{2}\)
\(=(-23)^{2}+(-23)^{2}+(-7)^{2}+(-4)^{2}+(-4)^{2}+5^{2}+6^{2}+10^{2}+18^{2}+22^{2}\)
\(=529+529 + 49+16+16+25+36+100+324+484=2108\)
\(s=\sqrt{\frac{2108}{9}}\approx\sqrt{234.22}\approx15.3\)

Step7: Calculate the inter - quartile range

The lower half of the data is \(48,48,64,67,67\), and the upper half is \(76,77,81,89,93\)
\(Q1\) (median of the lower half) \(=64\)
\(Q3\) (median of the upper half) \(=81\)
\(IQR = Q3-Q1=81 - 64 = 17\)

Step8: Fill in the 5 - number summary

\(\text{Minimum}=48\), \(\text{Median}=71.5\)

Answer:

Measures of Center	Measures of Spread	5 - Number Summary
Median=\(71.5\)	Standard Deviation\(\approx15.3\)	Lower Quartile, \(Q1 = 64\)
Mode=\(48\) and \(67\)	Interquartile Range, \(IQR = 17\)	Median=\(71.5\)
		Upper Quartile, \(Q3 = 81\)
		Maximum=\(93\)