Descriptive statistics ppt
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This document discusses descriptive statistics and how to calculate them. It covers preparing data for analysis through coding and tabulation. It then defines four types of descriptive statistics: measures of central tendency like mean, median, and mode; measures of variability like range and standard deviation; measures of relative position like percentiles and z-scores; and measures of relationships like correlation coefficients. It provides formulas for calculating common descriptive statistics like the mean, standard deviation, and Pearson correlation.
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Descriptive statistics ppt
- 2. TOPICS DISCUSSED IN THIS CHAPTER • Preparing data for analysis • Types of descriptive statistics – Central tendency – Variation – Relative position – Relationships • Calculating descriptive statistics
- 3. PREPARING DATA FOR ANALYSIS • Issues – Scoring procedures – Tabulation and coding – Use of computers
- 4. SCORING PROCEDURES • Instructions – Standardized tests detail scoring instructions – Teacher-made tests require the delineation of scoring criteria and specific procedures • Types of items – Selected response items - easily and objectively scored – Open-ended items - difficult to score objectively with a single number as the result
- 5. TABULATION AND CODING • Tabulation is organizing data – Identifying all information relevant to the analysis – Separating groups and individuals within groups – Listing data in columns • Coding – Assigning names to variables • EX1 for pretest scores • SEX for gender • EX2 for posttest scores
- 6. TABULATION AND CODING • Reliability – Concerns with scoring by hand and entering data – Machine scoring • Advantages – Reliable scoring, tabulation, and analysis • Disadvantages – Use of selected response items, answering on scantrons
- 7. TABULATION AND CODING • Coding – Assigning identification numbers to subjects – Assigning codes to the values of non-numerical or categorical variables • Gender: 1=Female and 2=Male • Subjects: 1=English, 2=Math, 3=Science, etc. • Names: 001=Rahul, 002=Rajani, 003= Rita, … 256=Harish
- 8. COMPUTERIZED ANALYSIS • Need to learn how to calculate descriptive statistics by hand – Creates a conceptual base for understanding the nature of each statistic – Exemplifies the relationships among statistical elements of various procedures • Use of computerized software – SPSS-Windows – Other software packages
- 9. DESCRIPTIVE STATISTICS • Purpose – to describe or summarize data in a parsimonious manner • Four types – Central tendency – Variability – Relative position – Relationships
- 10. DESCRIPTIVE STATISTICS • Graphing data – a frequency polygon – Vertical axis represents the frequency with which a score occurs – Horizontal axis represents the scores themselves SCORE 9.0 8.0 7.0 6.0 5.0 4.0 3.0 SCORE Frequency 5 4 3 2 1 0 Std. Dev = 1.63 Mean = 6.0 N = 16.00
- 11. CENTRAL TENDENCY • Purpose – to represent the typical score attained by subjects • Three common measures – Mode – Median – Mean
- 12. CENTRAL TENDENCY • Mode – The most frequently occurring score – Appropriate for nominal data • Median – The score above and below which 50% of all scores lie (i.e., the mid-point) – Characteristics • Appropriate for ordinal scales • Doesn’t take into account the value of each and every score in the data
- 13. CENTRAL TENDENCY • Mean – The arithmetic average of all scores – Characteristics • Advantageous statistical properties • Affected by outlying scores • Most frequently used measure of central tendency – Formula
- 14. VARIABILITY • Purpose – to measure the extent to which scores are spread apart • Four measures – Range – Quartile deviation – Variance – Standard deviation
- 15. VARIABILITY • Range – The difference between the highest and lowest score in a data set – Characteristics • Unstable measure of variability • Rough, quick estimate
- 16. VARIABILITY • Quartile deviation – One-half the difference between the upper and lower quartiles in a distribution – Characteristic - appropriate when the median is being used
- 17. VARIABILITY • Variance – The average squared deviation of all scores around the mean – Characteristics • Many important statistical properties • Difficult to interpret due to “squared” metric – Formula
- 18. VARIABILITY • Standard deviation – The square root of the variance – Characteristics • Many important statistical properties • Relationship to properties of the normal curve • Easily interpreted – Formula
- 19. THE NORMAL CURVE • A bell shaped curve reflecting the distribution of many variables of interest to educators
- 20. THE NORMAL CURVE • Characteristics – Fifty-percent of the scores fall above the mean and fifty-percent fall below the mean – The mean, median, and mode are the same values – Most participants score near the mean; the further a score is from the mean the fewer the number of participants who attained that score – Specific numbers or percentages of scores fall between ±1 SD, ±2 SD, etc.
- 21. THE NORMAL CURVE • Properties – Proportions under the curve • ±1 SD = 68% • ±1.96 SD = 95% • ±2.58 SD = 99% – Cumulative proportions and percentiles
- 22. SKEWED DISTRIBUTIONS • Positive – many low scores and few high scores • Negative – few low scores and many high scores • Relationships between the mean, median, and mode – Positively skewed – mode is lowest, median is in the middle, and mean is highest – Negatively skewed – mean is lowest, median is in the middle, and mode is highest
- 23. MEASURES OF RELATIVE POSITION • Purpose – indicates where a score is in relation to all other scores in the distribution • Characteristics – Clear estimates of relative positions – Possible to compare students’ performances across two or more different tests provided the scores are based on the same group
- 24. MEASURES OF RELATIVE POSITION • Types – Percentile ranks – the percentage of scores that fall at or above a given score – Standard scores – a derived score based on how far a raw score is from a reference point in terms of standard deviation units • z score • T score • Stanine
- 25. MEASURES OF RELATIVE POSITION • z score – The deviation of a score from the mean in standard deviation units – The basic standard score from which all other standard scores are calculated – Characteristics • Mean = 0 • Standard deviation = 1 • Positive if the score is above the mean and negative if it is below the mean • Relationship with the area under the normal curve
- 26. MEASURES OF RELATIVE POSITION • z score (continued) – Possible to calculate relative standings like the percent better than a score, the percent falling between two scores, the percent falling between the mean and a score, etc. – Formula
- 27. MEASURES OF RELATIVE POSITION • T score – a transformation of a z score where T = 10(z) + 50 – Characteristics • Mean = 50 • Standard deviation = 10 • No negative scores
- 28. MEASURES OF RELATIVE POSITION • Stanine – a transformation of a z score where the stanine = 2(z) + 5 rounded to the nearest whole number – Characteristics • Nine groups with 1 the lowest and 9 the highest • Categorical interpretation • Frequently used in norming tables
- 29. MEASURES OF RELATIONSHIP • Purpose – to provide an indication of the relationship between two variables • Characteristics of correlation coefficients – Strength or magnitude – 0 to 1 – Direction – positive (+) or negative (-) • Types of correlation coefficients – dependent on the scales of measurement of the variables – Spearman rho – ranked data – Pearson r – interval or ratio data
- 30. MEASURES OF RELATIONSHIP • Interpretation – correlation does not mean causation • Formula for Pearson r
- 31. CALCULATING DESCRIPTIVE STATISTICS • Symbols used in statistical analysis • General rules for calculating by hand – Make the columns required by the formula – Label the sum of each column – Write the formula – Write the arithmetic equivalent of the problem – Solve the arithmetic problem
- 32. CALCULATING DESCRIPTIVE STATISTICS • Using SPSS Windows – Means, standard deviations, and standard scores • The DESCRIPTIVE procedures • Interpreting output – Correlations • The CORRELATION procedure • Interpreting output
- 34. FORMULA FOR THE MEAN n x X
- 35. FORMULA FOR VARIANCE 1 2 2 2 N N x S x x
- 36. FORMULA FOR STANDARD DEVIATION 1 2 2 N N x SD x
- 37. FORMULA FOR PEARSON CORRELATION N y y N x x N y x xy r 2 2 2 2
- 38. FORMULA FOR Z SCORE sx X x Z ) (