Session VI

“Measures of Central Tendency”

Measures of Central Tendency

The three measures in common use are:

Average

Median

Mode

Average

There are three different techniques available for

calculating the average

Ungrouped data

Grouped data

Weighted average

Average-Ungrouped Data

1

ni

i

XX

n

Xi=readings n=number of observations

Average-Grouped Data

1

1 1 2 2

1 2

... .

...

hi i

i

h h

h

f XX

n

f X f X f X

f f f

h = number of cells fi=frequencyXi=midpoint n=number of observations

Average-Weighted Average

1

1

n

i iiw n

i

i

w XX

w

Used when a number of averages are combined with different frequencies

Xi=readings n=number of observationsW=weight of average

Median-Grouped Data

2m

d m

m

ncf

M L if

lm = lower boundary of the cell with the median

n = total number of observations

cfm= cumulative frequency of all cells below m

fm = frequency of median cell

i =cell interval

Example

Boundaries Midpoint Frequency Computation

23.6-26.5 25.0 4 100

26.6-29.5 28.0 36 1008

29.6-32.5 31.0 51 1581

32.6-35.5 34.0 63 2142

35.6-38.5 37.0 58 2146

38.6-41.5 40.0 52 2080

41.6-44.5 43.0 34 1462

44.6-47.5 46.0 16 736

47.6-50.5 49.0 6 294

Total 320 11549

Frequency Distribution of the Life of 320 tires in 1000 km

Median-Grouped Data

2m

d m

m

ncf

M L if

320154

235.6 3 35.958

Md

Using data from example table

Mode

The Mode is the value that occurs with the

greatest frequency.

It is possible to have no modes in a series or

numbers or to have more than one mode.

Relationship Among theMeasures of Central Tendency

Measures of Dispersion

Range

Standard Deviation

Variance

Range

The range is the simplest and easiest to

calculate of the measures of dispersion.

Range = R = Xh – Xl

Largest value - Smallest value in data set

Standard Deviation

Sample Standard Deviation:

2

1( )

1

n

iXi X

Sn

Standard Deviation

Ungrouped Technique

2 2

1 1( )

( 1)

n n

i in Xi Xi

Sn n

2

2

11

/

1

nn

ii

Xi Xi n

Sn

Standard Deviation

2 2

11

( ) ( )

( 1)

hh

i i i ii

i

n f X f X

sn n

Grouped Technique

Variance

2

2

11

/

1

nn

ii

Xi Xi n

Sn

Comments on the Measures of Dispersion

As n increases, accuracy of R decreases

Use R when there is small amount of data or data is

too scattered

If n > 10 use standard deviation

A smaller standard deviation means better quality

Comparison of two distributions with equal average and range

Other Measures

There are three other measures that are

frequently used to analyze a collection of data:

Skewness

Kurtosis

Coefficient of Variation

Skewness is the lack of symmetry of the data.

For grouped data:

Skewness

3

13 3

( ) /h

i ii

f X X na

s

Skewness

Left (negative) and right (positive) skewness distributions

Kurtosis provides information regarding the shape of the

population distribution (the peakedness or heaviness of

the tails of a distribution).

For grouped data:

Kurtosis

4

14 4

( ) /h

i ii

f X X na

s

Kurtosis

Leptokurtic and Platykurtic distributions

Correlation variation (CV) is a measure of how much

variation exists in relation to the mean.

(100%)sCV

X

Coefficient of Variation

Population

Set of all items that possess a characteristic of

interest

Sample

Subset of a population

Population and Sample

Parameter is a characteristic of a population, it describes a

population

Example: average weight of the population, e.g. 50,000 cans

made in a month.

Statistic is a characteristic of a sample, used to make

inferences on the population parameters that are

typically unknown, called an estimator

Example: average weight of a sample of 500 cans from that

month’s output, an estimate of the average weight of

the 50,000 cans.

Parameter and Statistic

Comparison of Sample and Population

Statistic Parameter

X average μ(Xo) mean

s sample standard deviation σ(so) standard deviation