Question

The following variable (X) represents the number of coupons used over a 6 month period by a sample of 11 shoppers:

81, 64, 71, 68, 71, 55, 62, 71, 66, 77, 61.  

Use this data to compute:

The mean, the median, the mode, the range, the variance, the standard deviation, the 80th percentile, the sum of the values of (X), and the sum of the squared deviations of each value of (X) from the mean. In addition, please explain what information is provided to us about this variable by your answers to: (1) the sample mean and (2) the sample standard deviation.

Answer 1

The given observations-

81,64,71,68,71,55,62,71,66,77,61

Number of observation, n=11

Mean, = 747/11 = 67.90909= 67.91(approx.)

Arranging the observations in ascending order-

55,61,62,64,66,68,71,71,71,77,81

Median is given by the formula [(n+1)/2]th observation = 6th. observation = 68

Therefore, Median = 68

It is evident from the given observations that the Mode is 71(Highest no. of times this observation is repeated(3))

Therefore, Mode = 71

Range= (Highest value - Lowest value) from the given observation set

Highest Value=81, Lowest value=55

Range=81-55=26

The table below shows the calculations-

	Observations(Xi)	(Xi - )	(Xi - )2
	81	13.09	171.3481
	64	-3.91	15.2881
	71	3.09	9.5481
	68	0.09	.0081
	71	3.09	9.5481
	55	-12.91	166.6681
	62	-5.91	34.9281
	71	3.09	9.5481
	66	-1.91	3.6481
	77	9.09	82.6281
	61	-6.91	47.7481
Total	747	-0.01	550.9091

Arranging the observations in ascending order-

55,61,62,64,66,68,71,71,71,77,81

The 80th percentile of the observation set is [(80/100)x (n+1)]th observation = 10th observation which equals 77

Answer 1:

The Sample mean equals the original mean, that is, 67.91

Answer 2:

The Sample Standard deviation equals {[ (X_i - )² ] / n(n-1)}^1/2 = 2.2379 = 2.24

All the answers-

• Mean = 67.91
• Median= 68
• Mode=71
• Range = 26
• Variance=50.08
• Standard Deviation=7.08
• The 80th Percentile=77
• The sum of the values of X = 747
• The sum of the squared deviations of each value of X from the mean=550.91
• The Sample mean=67.91
• The Sample standard deviation=2.24

(Note that all the answers are rounded upto 2 decimal places)

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Variance is given by the formula Observation] Im- no - 2(x - 5)? I (550.909) = 50.0826 ~ 50.08 Standard deviation & variance ~ 7.08

1) equals the original mean. The that sampe is, mean 67.91 27 The Sample Standard deviation = J + 2(x;-) 550.909) - 7.42