Question

question #8
find the interquartile range and sample standard deviation for the set of data.
single family home prices
339,400 345,000 332,600 341,500
325,200 318,500 358,000 321,100
323,700 333,800 351,000 337,700
355,100 339,400 352,000 341,500
337,700

o iqr = 18,300 and s = 12,299.9
o iqr = 11,000 and s = 8,850.79
o iqr = 22,900 and s = 12,466.55
o iqr = 19,100 and s = 11,760.5

Explanation:

Step1: Sort the data

First, sort the single - family home prices data in ascending order: 318500, 321100, 323700, 325200, 332600, 333800, 337700, 337700, 339400, 339400, 341500, 341500, 345000, 351000, 352000, 355100, 358000.

Step2: Find quartiles

There are \(n = 17\) data points. The median (\(Q_2\)) is the 9th value, so \(Q_2=339400\). The lower half of the data has 8 values. The median of the lower half (\(Q_1\)) is the average of the 4th and 5th ordered values. \(Q_1=\frac{325200 + 332600}{2}=328900\). The upper half of the data also has 8 values. The median of the upper half (\(Q_3\)) is the average of the 13th and 14th ordered values. \(Q_3=\frac{345000+351000}{2}=348000\).

Step3: Calculate IQR

The inter - quartile range \(IQR = Q_3 - Q_1=348000 - 328900 = 19100\).

Step4: Calculate sample mean

The sample mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}\). \(\sum_{i=1}^{17}x_i=318500+321100+323700+325200+332600+333800+337700+337700+339400+339400+341500+341500+345000+351000+352000+355100+358000 = 5773500\). \(\bar{x}=\frac{5773500}{17}\approx339617.65\).

Step5: Calculate sample standard deviation

The sample standard deviation formula is \(s=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}\).
\((x_1-\bar{x})^2=(318500 - 339617.65)^2\), \((x_2-\bar{x})^2=(321100 - 339617.65)^2\), etc. After calculating \(\sum_{i = 1}^{n}(x_i-\bar{x})^2\) and dividing by \(n - 1=16\) and taking the square - root, we get \(s\approx11760.5\).

Answer:

IQR = 19100 and s = 11760.5