Question

The following sample data reflect electricity bills for ten households in Istanbul in April 2020

$118.20	$67.88	$133.40	$88.42	$110.34
$76.90	$144.56	$127.89	$89.34	$129.10

Q3a.)

Determine three measures of central tendency for these sample data. Then, based on these measures, determine whether the sample data are symmetric or skewed.

Q3b.)

Compute the range, variance, and standard deviation for these sample data. Discuss which of these three measures you would prefer to use as a measure of variation.

Answer 1

	x	Arrange x	x -mean	(x -mean)2
	118.2	67.88	-40.723	1658.36273
	67.88	76.9	-31.703	1005.08021
	133.4	88.42	-20.183	407.353489
	88.42	89.34	-19.263	371.063169
	110.34	110.34	1.737	3.017169
	76.9	118.2	9.597	92.102409
	144.56	127.89	19.287	371.988369
	127.89	129.1	20.497	420.127009
	89.34	133.4	24.797	614.891209
n=	129.1	144.56	35.957	1292.90585
10	Sum=	1086.03		6236.89161
	Mean=	108.603		623.689161
	Median=	(5th +6th)/2
	Median=	(110.34+118.2)/2
	Median=	114.27
	3Median=	342.81
	2Mean=	217.206
	Mode =	3 Md-2Mean
	Mode =	125.604
	sample data are symmetric if
	Mean=	Median=	Mode
	Skewed positive if
	Mean  >	Median. >	Mode
	Skewed Negative if
	Mean  <	Median  <	Mode
	108.603.  <	114.27    <	125.604
	so sample data are negative skewed
	Range =	Max - Min
	Range =	144.56 -67.88
	Range =	76.68
	Variance=	Sum of  (x -mean)2 /n
	Variance=	623.689
	Standard deviation =sqrt (variance)
		24.97376624

A measure of variability is a summary statistic that represents the amount of variation in a dataset.

Measures of Variability: Range, Interquartile Range,Mean deviation , Variance, and Standard Deviation .

For best measure of dispersion we know that

It should be rigidly defined.

For the same data, all the methods should produce the same result.

It should be based on all the items so as to be more representative.

It should be amenable to further algebraic treatment.

It should have sampling stability

Range is not fit for the above properties

So

Standard deviation is considered to be the best measure of dispersion or variation and is thereore, the most widely used measure of dispersion.