Question

1. hospitals the following set of data represents the number of hospitals for selected states. find the mean, median, mode, midrange, range, variance, and standard deviation for th data. 53 84 28 78 35 111 40 166 108 60 123 87 84 74 80 62 source: world almanac.

Explanation:

Step1: Sort the data

Sort the data in ascending order: \(28,35,40,53,60,62,74,78,80,84,84,87,108,111,123,166\)

Step2: Calculate the mean

The mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\). Here \(n = 16\) and \(\sum_{i=1}^{16}x_{i}=28 + 35+40+53+60+62+74+78+80+84+84+87+108+111+123+166=1293\). So \(\bar{x}=\frac{1293}{16}=80.8125\)

Step3: Calculate the median

Since \(n = 16\) (even), the median \(M=\frac{x_{\frac{n}{2}}+x_{\frac{n}{2}+1}}{2}\). \(x_{8}=78\), \(x_{9}=80\). So \(M=\frac{78 + 80}{2}=79\)

Step4: Calculate the mode

The mode is the most - frequently occurring value. Here, the mode is \(84\)

Step5: Calculate the midrange

The midrange \(MR=\frac{\text{Min}+\text{Max}}{2}\). Min \(=28\), Max \(=166\). So \(MR=\frac{28+166}{2}=97\)

Step6: Calculate the range

The range \(R=\text{Max}-\text{Min}\). So \(R=166 - 28=138\)

Step7: Calculate the variance

The variance \(s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}\)
\(\sum_{i=1}^{16}(x_{i}-\bar{x})^{2}=(28 - 80.8125)^{2}+(35 - 80.8125)^{2}+\cdots+(62 - 80.8125)^{2}\)
\(=( - 52.8125)^{2}+(-45.8125)^{2}+\cdots+(-18.8125)^{2}\)
\(=2789.179688+2098.890625+\cdots+353.90625\)
\(=10930.875\)
\(s^{2}=\frac{10930.875}{15}\approx728.725\)

Step8: Calculate the standard deviation

The standard deviation \(s=\sqrt{s^{2}}\). So \(s=\sqrt{728.725}\approx26.995\)

Answer:

Mean: \(80.8125\), Median: \(79\), Mode: \(84\), Midrange: \(97\), Range: \(138\), Variance: \(728.725\), Standard deviation: \(26.995\)