Question

question #5
the population for 16 large us cities is listed in the table below. find the interquartile range and the standard deviation for the set of data.

city	population	city	population
minneapolis	382,578	seattle	608,660
winston - salem	229,617	lexington	295,803
cleveland	396,815	charlotte	731,424
madison	233,209	st. louis	319,294
omaha	408,958	columbus	787,033
chula vista	243,916	honolulu	337,256
milwaukee	594,833	austin	790,390

iqr = 478,937.5 and s = 893,382.17
iqr = 747,342 and s = 1,945,050.61
iqr = 337,254.5 and s = 204,065.32
saved iqr = 327,202 and s = 386,752.17

Explanation:

Step1: Sort the data

First, sort the population data in ascending order: 229573, 229617, 233209, 243916, 285068, 295803, 319294, 337256, 382578, 396815, 408958, 594833, 608660, 731424, 787033, 790390.

Step2: Find quartiles

Since \(n = 16\), the median (\(Q_2\)) is the average of the 8th and 9th - ordered values. \(Q_2=\frac{337256 + 382578}{2}=359917\). The lower half of the data is the first 8 values. The median of the lower - half (\(Q_1\)) is the average of the 4th and 5th ordered values in the lower half. \(Q_1=\frac{243916+285068}{2}=264492\). The upper half of the data is the last 8 values. The median of the upper - half (\(Q_3\)) is the average of the 4th and 5th ordered values in the upper half. \(Q_3=\frac{594833 + 608660}{2}=601746.5\).

Step3: Calculate IQR

The inter - quartile range \(IQR=Q_3 - Q_1=601746.5−264492 = 337254.5\).

Step4: Calculate the mean

The mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}=\frac{229573+229617+\cdots+790390}{16}=\frac{7177455}{16}=448590.9375\).

Step5: Calculate the standard deviation

The formula for the sample standard deviation \(s=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}\).
\((x_1-\bar{x})^2=(229573 - 448590.9375)^2\), \((x_2-\bar{x})^2=(229617 - 448590.9375)^2,\cdots,(x_{16}-\bar{x})^2=(790390 - 448590.9375)^2\).
\(\sum_{i = 1}^{16}(x_i-\bar{x})^2 = 63879199797.5625\).
\(s=\sqrt{\frac{63879199797.5625}{15}}\approx204065.32\).

Answer:

IQR = 337254.5 and s = 204065.32