Question

A survey asked people for their height in inches. Their responses were: 60, 74, 66, 70, 58, 63, 67, 71, 76, 65, 75, 72, 68, 65, 64, 68, 70, 72, 71, 64, 59, 77, 66, 65

a. Calculate the measures of central tendency

b. Give the five number summary of this data.

c. What is the range?

d. What is the standard deviation rounded to the nearest whole number?

Answer 1

(a) measures of central tendency

Mean : mean = sum of terms / number of terms

mean = 271 / 4

= 67.75

Median :

the median is the middle number in a sorted list of numbers. So, to find the median, we need to place the numbers in value order and find the middle number.

Ordering the data from least to greatest, we get:

58   59   60   63   64   64   65   65   65   66   66   67   68   68   70   70   71   71   72   72   74   75   76   77   

As you can see, we do not have just one middle number but we have a pair of middle numbers, so the median is the average of these two numbers:

Median = (67 + 68 ) / 2 = 67.5

Mode :

The mode of a set of data is the value in the set that occurs most often.

We see that the mode is 65

( b) five number summary

minimum : 58

first quartile :

first quartile (or lower quartile or 25th percentile) is the median of the bottom half of the numbers. So, to find the first quartile, we need to place the numbers in value order and find the bottom half.

58   59   60   63   64   64   65   65   65   66   66   67   68   68   70   70   71   71   72   72   74   75   76   77   

So, the bottom half is

58   59   60   63   64   64   65   65   65   66   66   67   

The median of these numbers is 64.5.

first quartile : 64.5.

Median :

median is the middle number in a sorted list of numbers. So, to find the median, we need to place the numbers in value order and find the middle number.

Ordering the data from least to greatest, we get:

58   59   60   63   64   64   65   65   65   66   66   67   68   68   70   70   71   71   72   72   74   75   76   77   

As you can see, we do not have just one middle number but we have a pair of middle numbers, so the median is the average of these two numbers:

Median = (67 + 68 ) / 2 = 67.5

Third quartile;

third quartile (or upper quartile or 75th percentile) is the median of the upper half of the numbers. So, to find the third quartile, we need to place the numbers in value order and find the upper half.

58   59   60   63   64   64   65   65   65   66   66   67   68   68   70   70   71   71   72   72   74   75   76   77   

So, the upper half is

68   68   70   70   71   71   72   72   74   75   76   77   

The median of these numbers is 71.5.

Third quartile: 71.5

Maxumum : 77

( C ) Range = Maximum - minimum

= 77 - 58

= 19

( d) standard deviation:

data	data-mean	(data - mean)2
60	-7.75	60.0625
74	6.25	39.0625
66	-1.75	3.0625
70	2.25	5.0625
58	-9.75	95.0625
63	-4.75	22.5625
67	-0.75	0.5625
71	3.25	10.5625
76	8.25	68.0625
65	-2.75	7.5625
75	7.25	52.5625
72	4.25	18.0625
68	0.25	0.0625
65	-2.75	7.5625
64	-3.75	14.0625
68	0.25	0.0625
70	2.25	5.0625
72	4.25	18.0625
71	3.25	10.5625
64	-3.75	14.0625
59	-8.75	76.5625
77	9.25	85.5625
66	-1.75	3.0625
65	-2.75	7.5625

So, standard deviation = 5