Data Analysis and Presentation

1. Data Analysis and Presentation By TemtimAssefa solomonmolla9@gmail.com

2. Data Type • Quantitative data is classified as categorical and numerical data • Categorical data refer to data whose values cannot be measured numerically but can be either classified into sets (categories) such as sex (male and female), religion, department • Numerical data, which are sometimes termed ‘quantifiable’, are those whose values are measured or counted numerically as quantities • These are analyzed by different techniques

3. Quantitative Data Analysis • Two common types analysis • Descriptive statistics • to describe, summarize, or explain a given set of data • Inferential statistics • use statistics computed from a sample to infer about the population • It is concerned by making inferences from the samples about the populations from which they have been drawn

4. Common data analysis technique • Frequency distribution • Measures of central tendency • Measures of dispersion • Correlation • Regression • And more

5. Frequency distribution • It is simply a table in which the data are grouped into classes and the number of cases which fall in each class are recorded. • Shows the frequency of occurrence of different values of a single Phenomenon. • Main purpose • To facilitate the analysis of data. • To estimate frequencies of the unknown population distribution from the distribution of sample data and • To facilitate the computation of various statistical measures

6. Example – Frequency Distribution • In a survey of 30 organizations, the number of computers registered in each organizations is given in the following table • This data has meaning unless it is summarized in some form

7. Example The following table shows frequency distribution Number of computers

8. Example … • The above table can tell us meaningful information such as • How many computers most organizations has? • How many organizations do not have computers? • How many organizations have more than five computers? • Why the computer distribution is not the same in all organizations? • And other questions

9. Continuous frequency distribution • Continuous frequency distribution constructed when the values do not have discrete values like number of computers • Example is age, salary variables have continuous values

10. Constructing frequency table • The number of classes should preferably be between 5 and 20. However there is no rigidity about it. • As far as possible one should avoid values of class intervals as 3,7,11,26….etc. preferably one should have class intervals of either five or multiples of 5 like 10,20,25,100 etc. • The starting point i.e the lower limit of the first class, should either be zero or 5 or multiple of 5. • To ensure continuity and to get correct class interval we should adopt “exclusive” method. • Wherever possible, it is desirable to use class interval of equal sizes.

11. Constructing … You can create a frequency table with two variables This is called Bivariate frequency table

12. Graphs • You can plot your frequency distribution using bar graph, pie chart, frequency polygon and other type of charts • Computer Import in Ethiopia in 2010

13. Bar graph

14. Measures of central tendency • Mode shows values that occurs most frequently • is the only measure of central tendency that can be interpreted sensibly • Median is used to identify the mid point of the data

15. Central Tendency …. • Mean is a measure of central tendency • includes all data values in its calculation Mean = sum of observation (sum)/ Total no. of observation (frequency ) • The mean for grouped data is obtained from the following formula: • where x = the mid-point of individual class • f = the frequency of individual class • N = the sum of the frequencies or total frequencies.

16. Advantages of Mean • It should be rigidly defined. • It should be easy to understand and compute. • It should be based on all items in the data. • Its definition shall be in the form of a mathematical • formula. • It should be capable of further algebraic treatment. • It should have sampling stability. • It should be capable of being used in further statistical computations or processing • However affected by extreme data values in skewed distributions • For Skewed distribution, use median than mean

17. Exercise • Do the following exercise for the following IT staff data for 13 organizations named as O1 to O13 • 25, 18, 20, 10, 8, 30, 42, 20, 53, 25, 10, 20, 42 • What is the mode? • What is the median? • What is the mean? • Change into frequency table? • Plot on bar graph? Pie chart? • What you interpret from the data?

18. Measures of Dispersion • The measure of central tendency serve to locate the center of the distribution, • Do not measure how the items are spread out on either side of the center. • This characteristic of a frequency distribution is commonly referred to as dispersion. • Small dispersion indicates high uniformity of the items, • Large dispersion indicates less uniformity. • Less variation or uniformity is a desirable characteristic

19. Type of measure of dispersion • There are two types • Absolute measure of dispersion • Relative measure of dispersion. • Absolute measure of dispersion indicates the amount of variation in a set of values in terms of units of observations. For example, if computers measured by numbers, it shows dispersion by number • Relative measures of dispersion are free from the units of measurements of the observations. You may measure dispersion by percentage • Range is an absolute measure while coefficient of variation is the relative measure

20. Dispersion … • There are different type of dispersion measures • We look at Standard Deviation and Coefficient of variation • Karl Pearson introduced the concept of standard deviation in 1893 • Standard deviation is most frequently used one • The reason is that it is the square–root of the mean of the squared deviation • Square of standard deviation is called Variance

21. Standard Deviation • It is given by the formula • Calculate the standard deviation from the following data. • 14, 22, 9, 15, 20, 17, 12, 11 • The Answer is 4.18 or

22. Interpretation • We expect about two-thirds of the scores in a sample to lie within one standard deviation of the mean. • Generally, most of the scores in a normal distribution cluster fairly close to the mean, • There are fewer and fewer scores as you move away from the mean in either direction. • In a normal distribution, 68.26% of the scores fall within one standard deviation of the mean, • 95.44% fall within two standard deviations, and • 99.73% fall within three standard deviations.

23. Advantage of SD • Assume the mean is 10.0, and standard deviation is 3.36. • one standard deviation above the mean is 13.36 and one standard deviation below the mean is 6.64. • The standard deviation takes account of all of the scores and provides a sensitive measure of dispersion. • it also has the advantage that it describes the spread of scores in a normal distribution with great precision. • The most obvious disadvantage of the standard deviation is that it is much harder to work out than the other measures of dispersion like rank and percentiles

24. Coefficient of Variation • The Standard deviation is an absolute measure of dispersion. • However, It may not always applicable • The standard deviation of number of computers cannot be compared with the standard deviation of computer use osstudents, as both are expressed in different units, • standard deviation must be converted into a relative measure of dispersion for the purpose of comparison -- coefficient of variation • The is obtained by dividing the standard deviation by the mean and multiply it by 100 coefficient of variation = X 100

25. Skewness • skewness means ‘ lack of symmetry’ . • We study skewness to have an idea about the shape of the curve which we can draw with the help of the given data. • If in a distribution mean = median =mode, then that distribution is known as symmetrical distribution. • The spread of the frequencies is the same on both sides of the center point of the curve.

26. Symmetrical distribution • Mean = Median = Mode

27. Positively skewed distribution Negatively skewed distribution

28. Measures of Skewness • Karl – Pearason’ s coefficient of skewness • Bowley’ s coefficient of skewness • Measure of skewness based on moments We see Karl- Pearson, read others from the textbook • Karl – Pearson is the absolute measure of skewness = mean – mode. • Not suitable for different unit of measures • Use relative measure of skewness -- Karl – Pearson’ s coefficient of skewness, i.e (Mean –Mode)/standard deviation In case of ill defined mode, we use 3(Mean –median)/standard deviation

29. Kurtosis • All the frequency curves expose different degrees of flatness or peakedness – called kurtosis • Measure of kurtosis tell us the extent to which a distribution is more peaked or more flat topped than the normal curve, which is symmetrical and bell-shaped, is designated as Mesokurtic. • If a curve is relatively more narrow and peaked at the top, it is designated as Leptokurtic. • If the frequency curve is more flat than normal curve, it is designated as platykurtic.

30. Interpretation • Real word things are usually have a normal distribution pattern – Bell shape

31. Normal dist… • This implies that • 68% of the population is in side 1 • 95% of the population is inside 2 • 99% of the population is 3 • So you need to select a confidence limit to say your sample is statistically significant or not • For example, if more than 5% of the population falls outside 2 standard deviation, the difference between two groups of population is not statistically significant

32. Correlation • Correlation is used to measure the linear association between two variables • For example, assume X is IT skill and Y is IT use. Is there association b/n these two variables

33. Correlation … • Correlation expresses the inter-dependence of two sets of variables upon each other. • One variable may be called as independent variable (IV) and the other is dependent variable (DV) • A change in the IV has an influence in changing the value of dependent variable • For example IT use will increase organization productivity because have better information access and improve their skills and knowledge

34. Correlation Lines

35. Correlation Lines Perfect Correlation No Correlation

36. Type of Correlation • Simple • Multiple correlation • Partial correlation • In simple correlation, we study only two variables. • For example, number of computers and organization efficiency • In multiple correlation we study more than two variables simultaneously. • For example, usefulness and easy of use and IT adoption • In Partial and total correlation, it refers to the study of two variables excluding some other variable

37. Karl pearson’ s coefficient of correlation • Karl pearson, a great biometrician and statistician, suggested a mathematical method for measuring the magnitude of linear relationship between the two variables • Karl pearson’ s coefficient of correlation is the most widely used method of correlation where X = x - x , Y = y - y

38. Exercise Calculate the correlation for the following given data

39. Spear Man Rank Correlation • Developed by Edward Spearman in 1904 • It is studied when no assumption about the parameters of the population is made. • This method is based on ranks • It is useful to study the qualitative measure of attributes like honesty, colour, beauty, intelligence, character, morality etc. • The individuals in the group can be arranged in order and there on, obtaining for each individual a number showing his/her rank in the group

40. Formula • Where D2 = sum of squares of differences between the pairs of ranks. • n = number of pairs of observations. • The value of r lies between –1 and +1. If r = +1, there is complete agreement in order of ranks and the direction of ranks is also same. If r = -1, then there is complete disagreement in order of ranks and they are in opposite directions.

41. Advantage of Correlation • It is a simplest and attractive method of finding the nature of correlation between the two variables. • It is a non-mathematical method of studying correlation. It is easy to understand. • It is not affected by extreme items. • It is the first step in finding out the relation between the two variables. • We can have a rough idea at a glance whether it is a positive correlation or negative correlation. • But we cannot get the exact degree or correlation between the two variables

42. The Pearson Chi-square • it is the most common coefficient of association, which is calculated to assess the significance of the relationship between categorical variables. • It is used to test the null hypothesis that observations are independent of each other. • It is computed as the difference between observed frequencies shown in the cells of cross-tabulation and expected frequencies that would be obtained if variables were truly independent.

43. Chi-square … Where O is observed value E is expected value X2 is the association Where is X2 value and its significance level depend on the total number of observations and the number of cells in the table

44. Regression • Regression is used to estimate (predict) the value of one variable given the value of another. • The variable predicted on the basis of other variables is called the “dependent” or the ‘ explained’ variable and the other the ‘ independent’ or the ‘ predicting’ variable. • The prediction is based on average relationship derived statistically by regression analysis. • For example, if we know that advertising and sales are correlated we may find out expected amount of sales f or a given advertising expenditure or the required amount of expenditure for attaining a given amount of sales.

45. Regression • Regression is the measure of the average relationship between two or more variables in terms of the original units of the data. • Type of regression • Simple and Multiple • Linear and Non –Linear • Total and Partial

46. Simple and Multiple: • In case of simple relationship only two variables are considered, for example, the influence of advertising expenditure on sales turnover. • In the case of multiple relationship, more than two variables are involved. On this while one variable is a dependent variable the remaining variables are independent ones. • For example, the turnover (y) may depend on advertising expenditure (x) and the income of the people (z). • Then the functional relationship can be expressed as y = f (x,z).

47. Linear and Non-linear • The linear relationships are based on straight-line trend, the equation of which has no-power higher than one. But, remember a linear relationship can be both simple and multiple. • Normally a linear relationship is taken into account because besides its simplicity, it has a better predictive value, a linear trend can be easily projected into the future. • In the case of non-linear relationship curved trend lines are derived. The equations of these are parabolic.

48. Total and Partial • In the case of total relationships all the important variables are considered. • Normally, they take the form of a multiple relationships because most economic and business phenomena are affected by multiplicity of cases. • In the case of partial relationship one or more variables are considered, but not all, thus excluding the influence of those not found relevant for a given purpose.

49. Regression analysis • The goal of regression analysis is to develop a regression equation from which we can predict one score on the basis of one or more other scores. • For example, it can be used to predict a job applicant's potential job performance on the basis of test scores and other factors

50. Linear regression equation • Linear regression equation of Y on X is Y = a + bX ……. (1) • And X on Y is X = a + bY……. (2) Where a, b are constants. • In a regression equation, y is the dependent variable or criterion variable, or outcome variable we would like to predict. • X represents the variable we are using to predict y; x is called the predictor variable.