Question

question #8
find the interquartile range and sample standard deviation for the set of data.
kentucky 21
vermont 33.1
north carolina 26.5
texas 25.5
connecticut 35.6
hawaii 29.6
wisconsin 25.7
district of columbia 48.5
north dakota 25.8
tennessee 23
wyoming 34.5
utah 28.5
arizona 25.6
massachusetts 38.2
illinois 30.6
iqr = 7.1 and s = 4.15
iqr = 11.3 and s = 6.06
iqr = 5.7 and s = 4.66
saved iqr = 8.9 and s = 7.03

Explanation:

Step1: Sort the data

First, sort the data values: \(21,23,25.5,25.6,25.7,25.8,26.5,28.5,29.6,30.6,33.1,34.5,35.6,38.2,48.5\)

Step2: Find quartiles

There are \(n = 15\) data points.
The first quartile \(Q_1\):
The position of \(Q_1=\frac{n + 1}{4}=\frac{15+1}{4}=4\)th value. So \(Q_1 = 25.6\)
The third quartile \(Q_3\):
The position of \(Q_3=\frac{3(n + 1)}{4}=\frac{3\times(15 + 1)}{4}=12\)th value. So \(Q_3=34.5\)
The inter - quartile range \(IQR=Q_3-Q_1\)
\(IQR=34.5 - 25.6=8.9\)

Step3: Calculate the sample mean \(\bar{x}\)

\(\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}=\frac{21+23+25.5+25.6+25.7+25.8+26.5+28.5+29.6+30.6+33.1+34.5+35.6+38.2+48.5}{15}\)
\(\sum_{i=1}^{15}x_i=21+23+25.5+25.6+25.7+25.8+26.5+28.5+29.6+30.6+33.1+34.5+35.6+38.2+48.5 = 448.2\)
\(\bar{x}=\frac{448.2}{15}=29.88\)

Step4: Calculate the sample standard deviation \(s\)

The formula for the sample standard deviation is \(s=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}\)
\((x_1-\bar{x})^2=(21 - 29.88)^2=(- 8.88)^2 = 78.8544\)
\((x_2-\bar{x})^2=(23 - 29.88)^2=(-6.88)^2 = 47.3344\)
\((x_3-\bar{x})^2=(25.5 - 29.88)^2=(-4.38)^2 = 19.1844\)
\((x_4-\bar{x})^2=(25.6 - 29.88)^2=(-4.28)^2 = 18.3184\)
\((x_5-\bar{x})^2=(25.7 - 29.88)^2=(-4.18)^2 = 17.4724\)
\((x_6-\bar{x})^2=(25.8 - 29.88)^2=(-4.08)^2 = 16.6464\)
\((x_7-\bar{x})^2=(26.5 - 29.88)^2=(-3.38)^2 = 11.4244\)
\((x_8-\bar{x})^2=(28.5 - 29.88)^2=(-1.38)^2 = 1.9044\)
\((x_9-\bar{x})^2=(29.6 - 29.88)^2=(-0.28)^2 = 0.0784\)
\((x_{10}-\bar{x})^2=(30.6 - 29.88)^2=(0.72)^2 = 0.5184\)
\((x_{11}-\bar{x})^2=(33.1 - 29.88)^2=(3.22)^2 = 10.3684\)
\((x_{12}-\bar{x})^2=(34.5 - 29.88)^2=(4.62)^2 = 21.3444\)
\((x_{13}-\bar{x})^2=(35.6 - 29.88)^2=(5.72)^2 = 32.7184\)
\((x_{14}-\bar{x})^2=(38.2 - 29.88)^2=(8.32)^2 = 69.2224\)
\((x_{15}-\bar{x})^2=(48.5 - 29.88)^2=(18.62)^2 = 346.7044\)
\(\sum_{i = 1}^{15}(x_i-\bar{x})^2=78.8544 + 47.3344+19.1844+18.3184+17.4724+16.6464+11.4244+1.9044+0.0784+0.5184+10.3684+21.3444+32.7184+69.2224+346.7044=708.42\)
\(s=\sqrt{\frac{708.42}{15 - 1}}=\sqrt{\frac{708.42}{14}}\approx\sqrt{50.6014}\approx7.03\)

Answer:

\(IQR = 8.9\) and \(s = 7.03\)