Q. 32 - Elementary Statistics

Question

The U.S National Oceanic and Atmosphere Administrations publishes temperature data in Climatography of United States. According to that document, the annual average maximum and minimum temperatures for selected cities in the United States are as provided on the Weissstats site. 

a. obtain the mean, median and mode(s) of the data. Determine which of these measures of center is best and explain your answer.

b. determine the range and sample standard deviation of the data.

c. find the five-number summary and interquartile range of the data. 

d. identify potential outliers, if any

e. obtain and interpret a boxplot.

Step-by-Step Solution

Answer

Part a. 

Maximum Temperature:

Mean \(=65.3268\)

Median \(=64\)

Mode \(=54.5, 55.9, 57.6, 59.8, 62.6, 63.6, 65.1, 67.4, 67.8, 70.6\)

Minimum Temperature:

 Mean \(=45.5197\)

Median \(=44.2\)

Mode \(=35.8, 39.8, 41.2, 43.2, 44.8, 47.4, 48.6, 52.5\)

Part b. 

Maximum Temperature:

Standard deviation \(=8.7228\)

Minimum Temperature:

Standard deviation \(=9.3426\)

Part c. 

Maximum Temperature:

Minimum value \(=47.6\)

Quartile \(1=58.4\)

Median \(=64\)

Quartile\(3 =71.1\)

Maximum value \(=85.5\)

Interquartile range \(12.7\)

Minimum Temperature:

Minimum value \(=29.3\)

Quartile \(1=39.8\)

Median \(=44.2\)

Quartile\(3 =51.4\)

Maximum value \(=74.2\)

Interquartile range \(11.6\)

Part d. 

Maximum Temperature:

No Outliers

Minimum Temperature:

\(69.1,70.2\) and \(74.2\) are outliers.

Part e. See the Box-Plot

Part a. Step 1.

Maximum Temperature

\(\bar{x}=\frac{47.6+48.7+49.6+...+84.5+84.7++85.5}{71}=\frac{4638.2}{71}\approx 65.3268\)

Median i.e. the center value in the data is as per below:

Median \(=64\)

The mode is the number which is coming maximum number of times which is \(2th\) given data

Mode \(=54.5, 55.9, 57.6, 59.8, 62.6, 63.6, 65.1, 67.4, 67.8, 70.6\)

Minimum Temperature

\(\bar{x}=\frac{29.3+30.1+30.5+...+69.1+70.2+74.2}{71}=\frac{3231.9}{71}\approx 45.5197\)

Median i.e. the center value in the data is as per below:

Median \(=44.2\)

The mode is the number which is coming maximum number of times which is \(2th\) given data

Mode\(=35.8, 39.8, 41.2, 43.2, 44.8, 47.4, 48.6, 52.5\)

Part b. Step 1. Explanation

Maximum Temperature

The standard deviation can be calculated as per below:

\(s=\sqrt{\frac{(47.6-65.3369)^{2}+...+(85.5-65.3369)^{2}}{71-1}}\approx 8.7228\)

Minimum Temperature

The standard deviation can be calculated as per below:

\(s=\sqrt{\frac{(29.3-45.5197)^{2}+...+(74.2-45.5197)^{2}}{71-1}}\approx 9.3426\)

Part c. Step 1. Explanation

Maximum Temperature:

Minimum value \(=47.6\)

Quartile \(1=58.4\)

Median \(=64\)

Quartile\(3 =71.1\)

Maximum value \(=85.5\)

Interquartile range \(=Q_{3}-Q_{1}=71.1-58.4=12.7\)

Minimum Temperature:

Minimum value \(=29.3\)

Quartile \(1=39.8\)

Median \(=44.2\)

Quartile\(3 =51.4\)

Maximum value \(=74.2\)

Interquartile range \(=Q_{3}-Q_{1}=51.4-39.8=11.6\)

Part d. Step 1. Explanation

Maximum Temperature:

Outliers are the data values which are \(1.5\) times more than the interquartile range above \(Q_{3}\) and below \(Q_{1}\).

\(Q_{3}+1.5(IQR)=71.1+1.5(12.7)=90.15\)

\(Q_{1}-1.5(IQR)=58.4-1.5(12.7)=39.35\)

It can be seen that there are no outliers in the data as all the values are between \(39.35, 90.15\)

Minimum Temperature:

Outliers are the data values which are \(1.5\) times more than the interquartile range above \(Q_{3}\) and below \(Q_{1}\).

\(Q_{3}+1.5(IQR)=51.4+1.5(11.6)=68.8\)

\(Q_{1}-1.5(IQR)=51.4-1.5(11.6)=22.4\)

\(69.1, 70.2\) and \(74.2\) are outliers, as they are between \((22.4,68.8)\)

Part e. Step 1. Explanation

Maximum Temperature:

Minimum Temperature: