May 06th, Chapter - 7 INFORMATION PRESENTATION 7.1 Statistical analysis 7.2 Presentation of data 7.3 Averages 7.4 Index numbers 7.5 Dispersion from the average

May 06th, – Statistical Analysis  statistical analysis is a management tool used in decision making  Statistics is the technique for comparing numbers and drawing conclusions from them  Two important factors are involved –using numbers to arrive at a solution –comparing numbers  A number in isolation provides very little information  To be able to compare numbers they must be measured in the same way, be in the same units and must refer to the same items

May 06th,  vital statistics: to record population details –as required by rulers for raising taxes and armies  Governments use it to plan social services  Within industry, statistics is used for competitive analysis  The numbers which the technique provides cannot lie, but they are open to misinterpretation (deaths in bed) (deaths in bed)  Companies also generate data, mainly for their own use. e.g. data on orders received, advertising effectiveness, and defect rates etc.

May 06th, – Presentation of data  The items which are measured are referred to as variables  There can be two types of variable, discrete and continuous  Tables –the position of columns relative to each other  Pictorial Presentation –It gets over the essential facts very quickly and creates an impact. –For larger amounts of data it is also a good way for indicating trends, which may not be easy to deduce from the table  Picture elements used in this representation can vary depending on the items being compared, for example people for employment.

May 06th, Types of Pictorial Presentation  Pictograms Fig 7.1 –Picture elements used in this representation can vary depending on the items being compared, for example people for employment –It is best used when whole items are being compared, each symbol representing a unit  Bar charts –It may consist of single bars, multiple bars or component bars –length of bar shows the size of the item being compared –the component bar chart ???? –Bar charts can be drawn in two dimensions or three dimensions –Legends can be added to bar charts (numbers can be placed next to bar) –not very good when many items are involved

May 06th, Types of Pictorial Presentation  Histograms –Special form of bar chart in which the areas under the rectangles (that make up the bars) represent the relative frequency of occurrence of the item. Usually the height of the bars determines the frequency  Pie charts –Showing subdivisions of the whole –Not possible to read off absolute values

May 06th, Graphs  A straight line with dependent and independent variables  Honesty must also be used in drawing graphs (scale should not be deceiving if two lines are drawn on the same graph for comparison) –Performance of two students A & B on different scales: A has 78, 76, 79, 81 & 82 B has 50, 61, 65, 68 & 72  Types of graphs include: –Logarithmic scale graphs –Strata or band graphs –Ogives –Frequency polygons –Lorenz curves

May 06th, Types of graphs  Logarithmic scale graphs –The logarithmic scale graph (ratio graph) shows the gradations as a logarithmic or ratio. These graphs are used to show relative changes in data  Strata or band graphs –Several curves are drawn on the same paper –actual values are by the difference between them  Ogives –Ogives are a graphical method for representing cumulative data; as ‘more than’ and ‘less than’ –Figure 7.8

May 06th, Types of graphs  Frequency polygons –A frequency polygon may be considered to be a graphical version of a histogram –Derived from a histogram by joining the midpoints of the class intervals Figure 7.9 –It is also conventional that the frequency polygon is extended a half cell interval at either end  Lorenz curves –In statistical analysis it is often found that a small proportion of items have the greatest influence. For example a small percentage of the population have the highest income in a country –This law of inequality can be shown graphically by a Lorenz curve, and it allows management attention to be focused on the few critical elements that have the greatest influence. Fig 7.10

May 06th, – Averages  Presenting a single number, referred to as the central tendency, to represent many numbers  There are several types of average –The arithmetic mean –The median –The mode –The geometric mean –The harmonic mean

May 06th, Types of average  The arithmetic mean –the total of the values divided by the total number of items –It is easy to calculate and that it takes account of all the numbers –It is affected by extreme values –When calculating the arithmetic mean of percentage figures; the weighted average must be obtained  The median –The median is the middle figure, placing the figures in an ascending or descending order –For even number of items, the arithmetic mean of the two central numbers is taken as being the median –It can be used even when the items cannot be expressed as a number; e.g. colours put in order –not affected by extreme values; the extreme numbers can change without having any effect on the median –not representative of the result; if the numbers are irregular and widely spread

May 06th, Types of average  The mode –The mode is the item that appears most often –It is not affected by extreme values –There can be several modes (multimode), or there may be no mode –Not suitable for calculations; can give large errors when dealing with widely spread and erratic numbers  The geometric mean –Items are multiplied together and the root of the result is taken, the base of the root being equal to that of the number of items –Mostly used, where the value of the quantity depends on its previous value –It cannot be used if any item is zero or a negative

May 06th, Types of average  The harmonic mean –Reciprocal of the mean of reciprocals of each individual item –For example; km/hr → distance per unit time  if a car travels three equal distances (say 100 km) at different speeds, then the average speed is found as the harmonic mean  but if the car traveled for equal intervals of time, (say 20 minutes) then the average speed is found as the arithmetic mean

May 06th, – Index Numbers  An index number is a special type of average which is used when comparing different types of items  It is an average of a group of items, measured over time, to show the change in the items  In compiling an index number it is important to decide on (1) the type of items to be compared, (2) the number of items and (3) the base year  Because when one is dealing with different types of items: a weighted average is used  Different types of index numbers can be calculated, depending on the type of average used (such as a geometric index or an arithmetic index)  Calculations can also be made using a variable base, especially if weights are changing rapidly

May 06th, – Dispersion from the average  Measure of the spread or dispersion gives some more info; for example, are some students performing better than others ?, and if so, by how much ?  Dispersion can be seen from tables or graphs, but they are often required to be represented by one or two numbers; different techniques by which this can be done are: –The range –Quartile deviation –Mean deviation –Standard deviation –Skew ness

May 06th, Types of Dispersion  The range –difference between the largest and smallest numbers being considered  Quartile deviation –A quartile divides the series of figures into four equal parts as the median was seen to divide it into two equal parts –The inter quartile range is the difference between the first and third quartile numbers and the quartile deviation is half the inter quartile range –It is easy to calculate and is not affected by extreme values –However, it in effect ignores half the values in the series and gives no indication of clustering

May 06th, Types of Dispersion  Mean deviation –The mean deviation is the average of all deviations from the arithmetic mean, signs being ignored –Since signs are ignored, it is not suitable for use in further mathematical analysis  Standard deviation –Most frequently used measure of dispersion from the average –Square the deviation from the mean (so eliminating signs), find their average, and then take the square root of the result –Variance is just the square of the standard deviation

May 06th, Types of Dispersion  Skewness –The distribution of numbers within a series usually lies unequally on either side of the middle –The distribution is said to be positively skewed if it is biased to low values (mean > median) and it has a negative skew if biased the other way –Skewness of the distribution gives a measure of the deviation between the mean, median and mode –Usually this skewness is stated in relative terms, to make comparisons between different series easier