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Chapter - 7Chapter - 7INFORMATION PRESENTATIONINFORMATION PRESENTATION

7.1 7.1 Statistical analysis Statistical analysis

7.2 7.2 Presentation of dataPresentation of data

7.3 7.3 AveragesAverages

7.4 7.4 Index numbersIndex numbers

7.5 7.5 Dispersion from the averageDispersion from the average

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7.1 – Statistical Analysis 7.1 – Statistical Analysis statistical analysis is a management tool used in statistical analysis is a management tool used in

decision making decision making Statistics is the technique for comparing Statistics is the technique for comparing

numbers and drawing conclusions from them numbers and drawing conclusions from them Two important factors are involved Two important factors are involved

– using numbers to arrive at a solutionusing numbers to arrive at a solution– comparing numbers comparing numbers

A number in isolation provides very little A number in isolation provides very little information information

To be able to compare numbers they must be To be able to compare numbers they must be measured in the same way, be in the same units measured in the same way, be in the same units and must refer to the same items and must refer to the same items

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vital statistics: to record population detailsvital statistics: to record population details– as required by rulers for raising taxes and armies as required by rulers for raising taxes and armies

Governments use it to plan social servicesGovernments use it to plan social services Within industry, statistics is used for competitive Within industry, statistics is used for competitive

analysisanalysis The numbers which the technique provides The numbers which the technique provides

cannot lie, but they are open to misinterpretationcannot lie, but they are open to misinterpretation (deaths in bed)(deaths in bed)

Companies also generate data, mainly for their Companies also generate data, mainly for their own use. e.g. data on orders received, own use. e.g. data on orders received, advertising effectiveness, and defect rates etc.advertising effectiveness, and defect rates etc.

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7.2 – Presentation of data 7.2 – Presentation of data The items which are measured are referred to as The items which are measured are referred to as

variables variables There can be two types of variable, discrete and There can be two types of variable, discrete and

continuous continuous TablesTables

– the position of columns relative to each other the position of columns relative to each other Pictorial PresentationPictorial Presentation

– It gets over the essential facts very quickly and creates an It gets over the essential facts very quickly and creates an impact. impact.

– For larger amounts of data it is also a good way for indicating For larger amounts of data it is also a good way for indicating trends, which may not be easy to deduce from the table trends, which may not be easy to deduce from the table

Picture elements used in this representation can vary Picture elements used in this representation can vary depending on the items being compared, for example depending on the items being compared, for example people for employment.people for employment.

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Types of Pictorial PresentationTypes of Pictorial Presentation Pictograms Pictograms Fig 7.1Fig 7.1

– Picture elements used in this representation can vary depending Picture elements used in this representation can vary depending on the items being compared, for example people for on the items being compared, for example people for employmentemployment

– It is best used when whole items are being compared, each It is best used when whole items are being compared, each symbol representing a unitsymbol representing a unit

Bar chartsBar charts– It may consist of single bars, multiple bars or component bars It may consist of single bars, multiple bars or component bars – length of bar shows the size of the item being compared length of bar shows the size of the item being compared – the component bar chart the component bar chart ????????– Bar charts can be drawn in two dimensions or three dimensions Bar charts can be drawn in two dimensions or three dimensions – Legends can be added to bar charts (numbers can be placed Legends can be added to bar charts (numbers can be placed

next to bar)next to bar)– not very good when many items are involved not very good when many items are involved

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Types of Pictorial PresentationTypes of Pictorial Presentation

Histograms Histograms – Special form of bar chart in which the areas Special form of bar chart in which the areas

under the rectangles (that make up the bars) under the rectangles (that make up the bars) represent the relative frequency of occurrence represent the relative frequency of occurrence of the item. Usually the height of the bars of the item. Usually the height of the bars determines the frequency determines the frequency

Pie charts Pie charts – Showing subdivisions of the whole Showing subdivisions of the whole – Not possible to read off absolute valuesNot possible to read off absolute values

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GraphsGraphs A straight line with dependent and independent variablesA straight line with dependent and independent variables Honesty must also be used in drawing graphs (scale Honesty must also be used in drawing graphs (scale

should not be deceiving if two lines are drawn on the should not be deceiving if two lines are drawn on the same graph for comparison)same graph for comparison)– Performance of two students A & B on different scales:Performance of two students A & B on different scales:

A has 78, 76, 79, 81 & 82A has 78, 76, 79, 81 & 82B has 50, 61, 65, 68 & 72B has 50, 61, 65, 68 & 72

Types of graphs include:Types of graphs include:– Logarithmic scale graphs Logarithmic scale graphs – Strata or band graphs Strata or band graphs – Ogives Ogives – Frequency polygons Frequency polygons – Lorenz curves Lorenz curves

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Types of graphsTypes of graphs Logarithmic scale graphs Logarithmic scale graphs

– The logarithmic scale graph (ratio graph) shows the The logarithmic scale graph (ratio graph) shows the gradations as a logarithmic or ratio. These graphs gradations as a logarithmic or ratio. These graphs are used to show relative changes in data are used to show relative changes in data

Strata or band graphs Strata or band graphs – Several curves are drawn on the same paperSeveral curves are drawn on the same paper– actual values are by the difference between themactual values are by the difference between them

Ogives Ogives – Ogives are a graphical method for representing Ogives are a graphical method for representing

cumulative data; as ‘more than’ and ‘less than’ cumulative data; as ‘more than’ and ‘less than’ – Figure 7.8 Figure 7.8

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Types of graphsTypes of graphs

Frequency polygons Frequency polygons – A frequency polygon may be considered to be a A frequency polygon may be considered to be a

graphical version of a histogram graphical version of a histogram – Derived from a histogram by joining the midpoints of Derived from a histogram by joining the midpoints of

the class intervals the class intervals Figure 7.9 Figure 7.9 – It is also conventional that the frequency polygon is It is also conventional that the frequency polygon is

extended a half cell interval at either endextended a half cell interval at either end Lorenz curves Lorenz curves

– In statistical analysis it is often found that a small In statistical analysis it is often found that a small proportion of items have the greatest influence. For proportion of items have the greatest influence. For example a small percentage of the population have example a small percentage of the population have the highest income in a countrythe highest income in a country

– This law of inequality can be shown graphically by a This law of inequality can be shown graphically by a Lorenz curve, and it allows management attention to Lorenz curve, and it allows management attention to be focused on the few critical elements that have the be focused on the few critical elements that have the greatest influence. greatest influence. Fig 7.10Fig 7.10

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7.3 – Averages 7.3 – Averages Presenting a single number, referred to as Presenting a single number, referred to as

the the central tendencycentral tendency, to represent many , to represent many numbers numbers

There are several types of average There are several types of average – The arithmetic mean The arithmetic mean – The median The median – The mode The mode – The geometric mean The geometric mean – The harmonic mean The harmonic mean

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Types of averageTypes of average The arithmetic mean The arithmetic mean

– the total of the values the total of the values divided bydivided by the total number of items the total number of items – It is easy to calculate and that it takes account of all the numbers It is easy to calculate and that it takes account of all the numbers – It is affected by extreme values It is affected by extreme values – When calculating the arithmetic mean of percentage figures; the When calculating the arithmetic mean of percentage figures; the

weighted average must be obtained weighted average must be obtained The median The median

– The median is the The median is the middle figuremiddle figure, placing the figures in an , placing the figures in an ascending or descending orderascending or descending order

– For even number of items, the arithmetic mean of the two central For even number of items, the arithmetic mean of the two central numbers is taken as being the median numbers is taken as being the median

– It can be used even when the items cannot be expressed as a It can be used even when the items cannot be expressed as a number; e.g. colours put in ordernumber; e.g. colours put in order

– not affected by extreme values; the extreme numbers can not affected by extreme values; the extreme numbers can change without having any effect on the median change without having any effect on the median

– not representative of the result; if the numbers are irregular and not representative of the result; if the numbers are irregular and widely spread widely spread

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Types of averageTypes of average

The mode The mode – The mode is the item that The mode is the item that appears mostappears most often often – It is not affected by extreme valuesIt is not affected by extreme values– There can be several modes (multimode), or there There can be several modes (multimode), or there

may be no modemay be no mode– Not suitable for calculations; can give large errors Not suitable for calculations; can give large errors

when dealing with widely spread and erratic numberswhen dealing with widely spread and erratic numbers The geometric mean The geometric mean

– Items are multiplied together and the root of the result Items are multiplied together and the root of the result is taken, the base of the root being equal to that of the is taken, the base of the root being equal to that of the number of items number of items

– Mostly used, where the value of the quantity depends Mostly used, where the value of the quantity depends on its previous valueon its previous value

– It cannot be used if any item is zero or a negative It cannot be used if any item is zero or a negative

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Types of averageTypes of average

The harmonic mean The harmonic mean – Reciprocal of the Reciprocal of the mean of reciprocalsmean of reciprocals of each of each

individual item individual item – For example; km/hr For example; km/hr → distance per unit time→ distance per unit time

if a car travels if a car travels three equal distancesthree equal distances (say 100 km) (say 100 km) at different speeds, then the average speed is at different speeds, then the average speed is found as the harmonic meanfound as the harmonic mean

but if the car traveled for but if the car traveled for equal intervals of timeequal intervals of time, , (say 20 minutes) then the average speed is found (say 20 minutes) then the average speed is found as the arithmetic meanas the arithmetic mean

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7.4 – Index Numbers 7.4 – Index Numbers An index number is a special type of average which is An index number is a special type of average which is

used when comparing different types of items used when comparing different types of items It is an average of a group of items, measured over time, It is an average of a group of items, measured over time,

to show the change in the items to show the change in the items In compiling an index number it is important to decide on In compiling an index number it is important to decide on

(1) the type of items to be compared, (2) the number of (1) the type of items to be compared, (2) the number of items and (3) the base yearitems and (3) the base year

Because when one is dealing with different types of Because when one is dealing with different types of items: a weighted average is used items: a weighted average is used

Different types of index numbers can be calculated, Different types of index numbers can be calculated, depending on the type of average used (such as a depending on the type of average used (such as a geometric index or an arithmetic index)geometric index or an arithmetic index)

Calculations can also be made using a variable base, Calculations can also be made using a variable base, especially if weights are changing rapidly especially if weights are changing rapidly

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7.5 – Dispersion from the average7.5 – Dispersion from the average Measure of the spread or dispersion gives some Measure of the spread or dispersion gives some

more info; for example, are some students more info; for example, are some students performing better than others ?, performing better than others ?, and if so, and if so,

by how much ? by how much ? Dispersion can be seen from tables or graphs, Dispersion can be seen from tables or graphs,

but they are often required to be represented by but they are often required to be represented by one or two numbers; different techniques by one or two numbers; different techniques by which this can be done are:which this can be done are:– The range The range – Quartile deviation Quartile deviation – Mean deviation Mean deviation – Standard deviation Standard deviation – Skew ness Skew ness

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Types of DispersionTypes of Dispersion The range The range

– difference between the largest and smallest numbers difference between the largest and smallest numbers being considered being considered

Quartile deviation Quartile deviation – A quartile divides the series of figures into A quartile divides the series of figures into four equal four equal

partsparts as the median was seen to divide it into two as the median was seen to divide it into two equal partsequal parts

– The inter quartile range is the difference between the The inter quartile range is the difference between the first and third quartile numbers and the quartile first and third quartile numbers and the quartile deviation is half the inter quartile range deviation is half the inter quartile range

– It is easy to calculate and is not affected by extreme It is easy to calculate and is not affected by extreme valuesvalues

– However, it in effect ignores half the values in the However, it in effect ignores half the values in the series and gives no indication of clusteringseries and gives no indication of clustering

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Types of DispersionTypes of Dispersion

Mean deviation Mean deviation – The mean deviation is the average of all deviations The mean deviation is the average of all deviations

from the arithmetic mean, from the arithmetic mean, signs being ignoredsigns being ignored– Since signs are ignored, it is not suitable for use in Since signs are ignored, it is not suitable for use in

further mathematical analysis further mathematical analysis

Standard deviation Standard deviation – Most frequently used measure of dispersion from the Most frequently used measure of dispersion from the

averageaverage– Square the deviation from the mean (so eliminating Square the deviation from the mean (so eliminating

signs), find their average, and then take the square signs), find their average, and then take the square root of the result root of the result

– Variance is just the square of the standard deviationVariance is just the square of the standard deviation

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Types of DispersionTypes of Dispersion

Skewness Skewness – The distribution of numbers within a series usually lies The distribution of numbers within a series usually lies

unequally on either side of the middle unequally on either side of the middle – The distribution is said to be positively skewed if it is The distribution is said to be positively skewed if it is

biased to low values (mean > median) and it has a biased to low values (mean > median) and it has a negative skew if biased the other waynegative skew if biased the other way

– Skewness of the distribution gives a measure of the Skewness of the distribution gives a measure of the deviation between the mean, median and modedeviation between the mean, median and mode

– Usually this skewness is stated in relative terms, to Usually this skewness is stated in relative terms, to make comparisons between different series easier make comparisons between different series easier