Basics of Statistical Analysis
- 1. BASICS OF STATISTICAL ANALYSIS DR REKHA SHARMA ASSOCIATE PROFESSOR UGC- HUMAN RESOURCE DEVELOPMENT CENTRE, RASHTRASANT TUKADOJI MAHARAJ NAGPUR UNIVERSITY, NAGPUR
- 2. STATISTICS •A branch of applied mathematics concerned with – collection and interpretation of quantitative data and – the use of probability theory to estimate population parameters concerned with treatment of quantitative information from groups of individuals.
- 3. DATA • The data collected from various sources through the use of different tools and techniques generally consists of numerical figures, ratings, narrations, responses to open-ended questions comprising a questionnaire or an interview schedule, quotations, field notes etc.
- 4. TYPES OF DATA QUALITATIVE DATA • Qualitative data are verbal or other symbolic materials. • The detailed descriptions of observed behaviours, people, situations and events • These data are gathered by a variety of methods and techniques such as: • (i) observation; (ii) interviews; (iii) questionnaires, opinionnaires and inventories; and (iv) recorded data from newspapers and other documents maintained by educational institutions, courts, clinics, government and non-government agencies etc • Provide data of qualitative nature in the form of emotions of respondents, their thoughts and experiences QUANTITATIVE DATA • obtained by using various tools and tests based on scale of measurement: nominal, ordinal, interval or ratio. • The quantitative data are either parametric or non-parametric • Parametric data undergo metric scale measurement • Non-parametric data are obtained by applying nominal or ordinal scales of measurement. These data are either coded, counted or ranked.
- 5. This Photo by Unknown Author is licensed under CC BY-SA
- 6. LEVELS OF MEASUREMENT This Photo by Unknown Author is licensed under CC BY-SA
- 7. STATISTICAL METHODS • •Statistical methods are extensively used in analysing quantitative data in educational research. • •These methods are classified as: • • Descriptive statistics • • Inferential statistics
- 8. DESCRIPTIVE STATISTICS • •Describes the characteristics of a sample. • •No conclusions are extended beyond this group. • •Method of organizing and summarizing data in Tables, graphs or numbers. • –Frequencies, percent, cumulative percent • –Measures of Central Tendency: Mean ,mode, median • –Measures of Dispersion: Range, SD, SE of mean, QD • –Normal Probability Curve, Skewness, Kurtosis • –Measures of Relative Positions: Sigma Score (Z)), T score, Percentile Rank • –Measures of Relationship: Product Moment Correlation, Rank order correlation
- 9. MEASURES OF DESCRIPTIVE STATISTICS (i) Central Tendency or Averages (ii) Variability or Dispersion (iii) Relative Position (iv) Relationship
- 10. MEASURES OF CENTRAL TENDENCY OR AVERAGES •We use averages to describe the characteristics of samples. •But in statistical language, use of the term 'average’ does not convey any specific meaning, because there are a number of types of averages, only one of which may be appropriate to use in describing given characteristics of a sample/group. •Three are mostly used in the analysis of educational data. These include: Mean, Median and Mode
- 11. THE MEAN • •The Mean of a distribution is the arithmetic average. • •It is perhaps the most familiar, most frequently used and well understood average. • •It is computed by dividing the sum of all the observations by the total number of observation.
- 12. SELECTION AND APPLICATION OF MEAN • •Mean is a more useful measure of central tendency than its other measures. • •Mean is an appropriate statistic when • •i) the observations or measures are distributed symmetrically about the centre of the distribution; • •ii) the most stable measure of central tendency is desired; • •iii) additional statistics are to be computed later.
- 13. THE MEDIAN •The Median is a point in an ordered arrangement of observations, above and below which one-half of the observations fall. •It is a measure of a position rather than size.
- 14. SELECTION AND APPLICATION OF MEDIAN Median is computed when i) the distribution of the observation measures is markedly skewed; Ii) there is not sufficient time to compute a Mean; Iii) an incomplete distribution is given; iv) we are interested in whether cases fall within the upper or lower halves of distribution and not particularly in how far they are from the central point.
- 15. THE MODE The mode is defined as the most frequently occurring observation in a distribution. It is located by inspection rather than by computation. If there is only one value which occurs a maximum number of times, then the distribution is said to have one mode or to be unimodal. In some distributions there may be more than one mode. A two mode distribution is bimodal; more than two, multimodal.
- 16. SELECTION AND APPLICATION OF MODE The mode is the most appropriate statistics when i) the quickest estimate of central tendency is wanted; ii) a very rough estimate of central tendency will suffice; iii) the purpose is to know the most typical case.
- 17. MEASURES OF VARIABILITY OR DISPERSION (i) Range (ii) Average Deviation (iii) Standard Deviation
- 18. THE RANGE The range is the simplest measure of variability. It is the difference between the highest and lowest observation or score. The range is used when: i) the data are too scant or too scattered; ii) only an idea of extreme scores or of total spread is wanted.
- 19. SELECTION AND APPLICATION OF RANGE Although the range has the advantage of being easily calculated, it has some limitations. I) The value of range is based on only two extreme scores in the total distribution and thus it does not provide us any information of the variation of many other scores of the distribution. II) it is not a stable statistics as its value can differ from sample to sample drawn from the same population
- 20. AVERAGE DEVIATION (AD) • The average deviation or AD is the mean of the deviations of all of the separate observations or scores in a series taken from their mean. • However, in averaging deviations to find average deviation (AD), the signs (+ and -) are not taken into consideration i.e. all the deviations, whether plus or minus, are treated as positive. • The formula for computing average deviation of an ungrouped data is:
- 21. SELECTION AND APPLICATION OF AVERAGE DEVIATION (AD) Average deviation is used when: i) It is desired to consider all deviations from the mean according to their size; ii) Extreme deviations would affect standard deviation unduly.
- 22. QUARTILE DEVIATION (Q) The quartile deviation or Q is one-half the scale distance between the 75th and 25th percentiles in a frequency distribution. The 25th percentile or Q, is the first quartile on the score scale, the point below which lie 25 per cent of the scores. The 75th percentile or Q, is the third quartile on the score scale, the point below which lie 75 per cent of the score
- 23. SELECTION AND APPLICATION OF QUARTILE DEVIATION 1) The mean is the measure of central tendency; 2) There are scattered or extreme scores which would influence the standard deviation disproportionately; 3) The concentration around the median, the middle 50 per cent of cases, is of primary interest
- 24. STANDARD DEVIATION (SD) • The standard deviation (SD) is the most general and stable measure of variability. • It is the positive square root of variance. • The average of the squared deviations of the measures or scores from their mean is known as variance which is generally denoted as sigma .
- 25. SELECTION AND APPLICATION OF STANDARD DEVIATION (a) Standard deviation is used when: i) the statistics having greatest stability is sought; ii) extreme deviations exercise a proportionally greater effect upon the variability; iii) co-efficient of correlation and other statistics are subsequently computed.
- 26. NORMAL PROBABILITY CURVE • One of the most common probability distribution in statistics is the normal distribution, commonly represented by the normal curve. • This mathematical and theoretical distribution describes the expected distribution of sample means and many other chance occurrences. • The standardized normal distribution is a purely theoretical probability distribution, but it is the most useful distribution in inferential statistics.
- 27. NORMAL PROBABILITY CURVE • The normal probability curve is based upon the law of probability or probable occurrence of certain events. • The 'probability' of a given event is defined as the expected frequency of occurrence of this event among events of a like sort.
- 28. NORMAL PROBABILITY CURVE • For example, the probability of an unbiased coin falling heads is half and that of a dice showing a three-sport is 1/6. • When any set of observations confirms to this mathematical form, it can be represented by a bell shaped curve with given characteristics: This Photo by Unknown Author is licensed under CC BY-NC-ND
- 29. CHARACTERISTICS OF NORMAL PROBABILITY CURVE • 1. The curve is symmetrical around its vertical axis. It implies that the size, shape and slope of the curve on one side of the curve is identical to that on the other side of the curve. • 2. The values of mean, mode and median computed for a distribution following this curve always coincide and have the same value. Source: https://www.google.com/url?sa=i&url=https%3A%2F%2Fwww.simplypsychology.org%2Fnormal- distribution.html&psig=AOvVaw21IW4oXJ3rsYNQWzkve2e9&ust=1619209734801000&source=images&cd =vfe&ved=0CAIQjRxqFwoTCND2_cfYkvACFQAAAAAdAAAAABAD
- 30. CHARACTERISTICS OF NORMAL PROBABILITY CURVE • 3. The curve has no boundaries in either direction, for the curve never touches the base line, no matter how far it is extended i.e. the curve is asymptotic and extends from - m (minus infinity) to + m (plus infinity). • 4. The data 'cluster' around the mean. • 5. The percentages in a given standard deviation are greatest around the Mean and decrease as one moves away from This Photo by Unknown author is licensed under CC BY-SA.
- 31. KURTOSIS This Photo by Unknown author is licensed under CC BY-NC.
- 32. SKEWNESS This Photo by Unknown author is licensed under CC BY-SA.
- 33. APPLICATIONS OF THE NORMAL CURVE (i) to find the percentage of cases, or the number when N is known, that fall between the mean and a given 0 distance from the mean; (ii) the percentage of the total area included between the mean and a given 0 distance from the mean. 2. The normal curve is used to convert a raw score into standard score. 3. Normal curve is useful in calculating the percentile rank of scores in a normal distribution. 4. For normalising a given frequency distribution, the normal curve is used. (i) to find the percentage of cases, or the number when N is known, that fall between the mean and a given 0 distance from the mean; (ii) the percentage of the total area included between the mean and a given 0 distance from the mean. 2. The normal curve is used to convert a raw score into standard score. 3. Normal curve is useful in calculating the percentile rank of scores in a normal distribution. 4. For normalising a given frequency distribution, the normal curve is used.
- 34. INFERENTIAL STATISTICS • •Statistical methods used to draw generalizations beyond the sample with a known degree of accuracy.
- 35. INFERENTIAL STATISTICS • •It is a decision to choose the right way to do analysis according to • – Parametric vs nonparametric • – Level of confidence • – Hypothesis • –Type of dependent /independent variables • – Number of groups, means • – Correlation, significance, associations, relationships • – Reliability and validity of scales
- 36. TYPES OF TESTS •Independent t test •T test •Paired t test •F test (ANOVA) •Pearson’s coefficient of Correlation •CR test, Z test Parametric tests •Chi Square(Unrelated Samples) •Median Test(Unrelated •Mann-Whitney U test Samples) •Sign Test (Related Samples) •Wilcoxon Matched Paired Signed Ranks Test (Related Samples) Non Parametric tests
- 37. HYPOTHESIS • •A hypothesis is a proposed explanation for the occurrence of the phenomena that a researcher formulates prior to conducting an experiment. • Hypothesis is an assumption that is made on the basis of some evidence. • A research hypothesis is a hypothesis that is used to test the relationship between two or more variables.
- 38. TYPES OF HYPOTHESIS • Null Hypothesis: States that there is no relationship between the two variables being studied. Denoted by H0 • Alternate Hypothesis: The alternative hypothesis states that there is a relationship between the two variables being studied. Denoted by H1 or HA • Non Directional Hypothesis: predicts that the independent variable will have an effect on the dependent variable, but the direction of the effect is not specified. • Directional Hypothesis: It predicts the nature of the effect of the independent variable on the dependent variable.
- 39. LEVEL OF STATISTICAL SIGNIFICANCE (ALPHA - ERROR) • Two types of error can occur. • Wrongly rejecting a true null hypothesis (HO) is called Type-I error. • The probability of this error is popularly known as P-Value. • The maximum P-value allowed in a study problem is called the level of significance or sometimes 𝛂 level.
- 40. P VALUE, SIGNIFICANCE VALUE Statistical Decision Null Hypothesis True False Rejected Type I Error ✔️ Not Rejected ✔️ Type II Error
- 41. P VALUE • In null hypothesis significance testing, the p-value is the probability of obtaining test results at least as extreme as the results actually observed, under the assumption that the null hypothesis is correct. • The P-value is kept at a low level, mostly less than 5% or P<0.05. • A result is called statistically significant when P<0.05 and highly significant when P<0.01. • The complement of this is 95% used in confidence interval (CI). • Significance level + Confidence Interval = 100
- 42. REFERENCES • IGNOU Study Material MFN 009 Research Methods and BioStatistics, MSc Dietetics and Food Service Management. http://www.ignouhelp.in/ignou-mfn-09- study-material/ • IGNOU Study Material MES-016 Educational Research, Data Analysis and Interpretation http://egyankosh.ac.in/handle/123456789/2863 • Statistics In Psychology And Education by Henry E. Garrett, Paragon International Publishers, New Delhi