Chapter 4 Measures of Location

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Quantitative Methods forBusiness and Management

Unit 4: Measures of Location


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INTRODUCTION is a Greek letter (pronounced

"sigma") and is used to denote the

summation of a number of terms.

X1 + X2 + X3 + X4

X3 + X4 + X5 + X6 + X7


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USE OF MEASURES OF

LOCATION The main measures of location are

the:

Mean

Median

Mode.


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USE OF MEASURES OF

LOCATION Descriptive Use

Comparison of Distributions


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MEANS Arithmetic Mean

The arithmetic mean of a set of

observations is the total sum of theobservations divided by the numberof observations.


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MEANS Example 1

Find the mean monthly rainfall inTown A from the twelve monthlyobservations given in Table

4.1:


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MEANS Example 2

Using this data, find themean number of days onwhich an employee is latein a month.


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MEANS Example 2


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MEANS Example 3


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MEAN Advantages andDisadvantages of the Arithmetic Mean

(a) Advantages (i) It is easy to calculate as the only information you need is

the sum of all the observations and the number of

observations. (ii) It is a well known statistic and it is easily manipulated to

calculate other useful statistical measures. (iii) It uses the values of all the observations.

(b) Disadvantages (i) A few extreme values can cause distortion which makes it

unrepresentative of the data set. (ii) When the data is discrete it may produce a value which

appears to be unrealistic, e.g. in Example 2, the mean numberof days on which an employee is late is 2.47.

(iii) It cannot be read from a graph.


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Weighte

dMean

A firm owns sixfactories at which the

basic weekly wagesare given in column 2

Find the mean basicwage earned byemployees of the firm.


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Weighte

dMean


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Geometric Mean

The geometric mean is seldom used outside ofspecialist applications. It is appropriate when

dealing with data such as that which showsexponential growth (that is where the rate ofgrowth depends on the value of the variableitself)


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Harmonic Mean

Harmonic MeanAnother measure of central tendency which is only occasionallyused is the harmonic mean.

It is most frequently employed for averaging speeds where thedistances for each section of the journey are equal.


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Harmonic Mean - ExampleAn aeroplane travels a distance of 900 miles. If it covers the firstthird and the last third of the trip at a speed of250 mph and themiddle third at a speed of300 mph, find the average speed.


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MEDIAN Definition

If a set of n observations is arranged

in order of size then, if n is odd, the median is the value of the

middle observation;

if n is even, the median is the value of

the arithmetic mean of the two middleobservations.


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MEDIAN


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Med

ian Example 1


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Med

ian Example 2


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Med

ian Example 3


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Med

ian Example 3


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Other Way


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Med

ian(a) Advantages (i) Its value is not distorted by extreme values, open-

ended classes or classes of irregular width. (ii) All the observations are used to order the data even

though only the middle one or two observations are usedin the calculation. (iii) It can be illustrated graphically in a very simple way.

(b) Disadvantages (i) In a grouped frequency distribution the value of the

median within the median class can only be an estimate,whether it is calculated or read from a graph. (ii) Although the median is easy to calculate it is difficult

to manipulate arithmetically. It is of little use incalculating other statistical measures.


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QUANTILESDefinitions

If a set of data is arranged in

ascending order of size, quantiles arethe values of the observations whichdivide the number of observationsinto a given number ofequal parts.


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QUANTILES


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Deciles and Percentiles


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QUANTILES Example 1


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QUANTILES Example 1


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QUANTILES Example 1


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QUANTILES Example 1


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MODE Definition

If the variable is discrete, the mode is

that value of the variable which occursmost frequently. This value can befound by ordering the observations orinspecting the simple frequency

distribution or its histogram.


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MODE As it is possible for several values of

the variable or several class intervalsto have the same frequency, a set of

data may have several modes. A set of observations with one mode

is called unimodal.

A set of observations with two modes

is called bimodal. A set of observations with more than

two modes is called multimodal.


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MODE Example 1


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MODE Example 1


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Continuous Variable


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Mode

(a) Advantages

It is not distorted by extreme values of theobservations.

It is easy to calculate.(b) Disadvantages

It cannot be used to calculate any furtherstatistic.

It may have more than one value (althoughthis feature helps to show the shape of thedistribution).


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Chapter 4 Measures of Location