STATISTICAL TOOLS NEEDED IN ANALYZING TEST RESULTS Prof. Yonardo Agustin Gabuyo

1. STATISTICAL TOOLSNEEDED IN ANALYZING TEST RESULTSProf. Yonardo Agustin Gabuyo

2. Statistics is a branch of science which deals with the collection, presentation, analysis and interpretation of quantitative data.

3. Branches of Statistics Descriptive statistics  methods concerned w/ collecting, describing, and analyzing a set of data without drawing conclusions (or inferences) about a large group

4. Inferential statistics  methods concerned with the analysis of a subset of data leading to predictions or inferences about the entire set of data or population.

5. Examples of Descriptive Statistics  Presenting the Philippine population by constructing a graph indicating the total number of Filipinos counted during the last census by age group and sex  The Department of Social Welfare and Development (DSWD) cited statistics showing an increase in the number of child abuse cases during the past five years.

6. A new milk formulation designed to improve the psychomotor development of infants was tested on randomly selected infants. Based on the results, it was concluded that the new milk formulation is effective in improving the psychomotor development of infants. Examples of Inferential StatisticsSource: Pilot Training Course on Teaching Basic Statistics by Statistical Research and Training Center Philippine Statistical Association , Inc.

7. Example Teacher Ron-nick gave a personality test measuring shyness to 25,000 students. What is the average degree of shyness and what is the degree to which the students differ in shyness are the concerns of _________ statistics. A. inferential B. graphic C. correlational D. descriptive

8. Example This is a type of statistics that give/s information about the sample being studied. a. Inferential and co-relational b. Inferential c. Descriptive d. Co relational

9. Inferential StatisticsSource: Pilot Training Course on Teaching Basic Statistics by Statistical Research and Training Center Philippine Statistical Association , Inc. Larger Set (N units/observations) Smaller Set (n units/observations) Inferences and Generalizations

10. Types of Variables

11. Qualitative variables variables that can be express in terms of properties, characteristics, or classification(non-numerical values).

12. Quantitative Variables  variables that can be express in terms of numerical values.a)Discrete- variables that can be express in terms of whole number.b)Continuous- variables that can be express in terms whole number, fraction or decimal number.

13. Levels of Measurement • Nominal  Numbers or symbols used to classify • Ordinal scale  Accounts for order; no indication of distance between positions • Interval scale  Equal intervals; no absolute zero • Ratio scale  Has absolute zero

14. Methods of Collecting Data • Objective Method • Subjective Method • Use of Existing Records

15. Methods of Presenting Data  Textual  Tabular  Graphical

16. Summary Measures Location Variation Skewness Kurtosis Percentile Quartile Decile Maximum Minimum Range Coefficient of Variation Variance Central Tendency Inter-quartile Range Mean Mode Standard Deviation Median

17. Measures of Location A Measure of Location summarizes a data set by giving a “typical value” within the range of the data values that describes its location relative to entire data set. Some Common Measures: • Minimum, Maximum • Central Tendency Percentiles, Deciles, Quartiles

18. Maximum and Minimum • Minimumis the smallest value in the data set, denoted as MIN. • Maximum is the largest value in the data set, denoted as MAX.

19. Measure of Central Tendency  A single value that is used to identify the “center” of the data • it is thought of as a typical value of the distribution • precise yet simple • most representative value of the data

20. Mean • Most common measure of the center • Also known as arithmetic average Population Mean Sample Mean

21. Properties of the Mean  may not be an actual observation in the data set.  can be applied in at least interval level.  easy to compute.  every observation contributes to the value of the mean.

22. Mean = 5 Properties of the Mean subgroup means can be combined to come up with a group mean  easily affected by extreme values 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14 Mean = 6

23. Median  Divides the observations into two equal parts. • If n is odd, the median is the middle number. • If n is even, the median is the average of the 2 middle numbers.  Sample median denoted as while population median is denoted as

24. Properties of a Median  may not be an actual observation in the data set  can be applied in at least ordinal level  a positional measure; not affected by extreme values 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14 Median = 5

25. Mode  the score/s that occurs most frequently  nominal average  computation of the mode for ungrouped or raw data 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 No Mode Mode = 9

26. Properties of a Mode  can be used for qualitative as well as quantitative data  may not be unique  not affected by extreme values  may not exist

27. Mean, Median & Mode Use the mean when:  sampling stability is desired  other measures are to be computed

28. Mean, Median & Mode Use the median when:  the exact midpoint of the distribution is desired  there are extreme observations

29. Mean, Median & Mode Use the mode when:  when the "typical" value is desired  when the dataset is measured on a nominal scale

30. Example Which measure(s) of central tendency is(are) most appropriate when the score distribution is skewed? A. Mode B. Mean and mode C. Median D. Mean

31. Example In one hundred-item test, what does Jay-R’s score of 70 mean? A. He surpassed 70 of his classmate in terms of score B. He surpassed 30 of his classmates in terms of score C. He got a score above mean D. He got 70 items correct

32. Example Which of the following measures is more affected by an extreme score? A. Semi- inter quartile range B. Median C. Mode D. Mean

33. Example The sum of all the scores in a distribution always equals a. The mean times the interval size b. The mean divided by the interval size c. The mean times N d. The mean divided by N

34. Example Teacher B is researching on family income distribution which is symmetrical. Which measure/s of central tendency will be most informative and appropriate? A. Mode B. Mean C. Median D. Mean and Median

35. Example What measure/s of central tendency does the number 16 represent in the following score distribution? 14,15,17,16,19,20,16,14,16? • Mode only • Mode and median c. Median only d. Mean and mode

36. Example What is the mean of this score distribution: 40, 42, 45, 48, 50, 52, 54, 68? a. 51.88 b. 50.88 c. 49.88 d. 68

37. Example Which is the correct about MEDIAN? a. It is measure of variability b. It is the most stable measure of central tendency c. It is the 50th percentile d. It is significantly affected by extreme values

38. Example Which measure(s) of central tendency can be determined by mere inspection? a. Median b. Mode c. Mean d. Mode and Median

39. Example Here is a score distribution: 98,93,93,93,90,88,88,85,85,85,86, 70,70,51,34,34,34,, 20,18,15,12,9,8,3,1. Which is a characteristics of the scores distribution? A. Bi-modal B. Tri-modal C. Skewed to the right D. No discernible pattern

40. Example Which is true of a bimodal score distribution? a. the group tested has two identical scores that appeared most. b. the scores are either high or low. c. the scores are high. d. the scores are low.

41. Example STUDY THE TABLE THEN ANSWER THE QUESTION: ScoresPercent of Students 0-59 2% 60-69 8% 70-79 39% 80-89 38% 90-100 13%

42. In which scores interval is the median? a. In the interval 80 to 89 b. In between the intervals of 60-69 and 70-79 c. In the interval 70-79 d. In the interval 60-69

43. How many percent of the students got a score below 70? a. 2% b. 8% c. 10% d. 39%

44. Percentiles  Numerical measures that give the relative position of a data value relative to the entire data set.  Percentage of the students in the reference group who fall below student’s raw score.

45. Divides the scores in the distribution into 100 equal parts (raw data arranged in increasing or decreasing order of magnitude).  The jth percentile, denoted as Pj, is the data value in the data set that separates the bottom j% of the data from the top (100-j)%.

46. EXAMPLE Suppose JM was told that relative to the other scores on a certain test, his score was the 97th percentile.  This means that 97% of those who took the test had scores less than JM’s score, while 3% had scores higher than JM’s.

47. Deciles Divides the scores in the distribution into ten equal parts, each part having ten percent of the distribution of the data values below the indicated decile.  The 1st decile is the 10th percentile; the 2nd decile is the 20th percentile…..  9th decile is the 90th percentile.

48. Quartiles  Divides the scores in the distribution into four equal parts, each part having 25% of the scores in the distribution of the data values below the indicated quartile.  The 1st quartile is the 25th percentile; the 2nd quartile is the 50th percentile, also the median and the 3rd quartile is the 75th percentile.

49. Example Robert Joseph’s raw score in the mathematics class is 45 which equal to 96th percentile. What does this mean? a. 96% of Robert Joseph’s classmates got a score higher than 45. b. 96% of Robert Joseph’s classmates got a score lower than 45. c. Robert Joseph’s score is less than 45% of his classmates. d. Roberts Joseph’s is higher than 96% of his classmates.

50. Example Which one describes the percentile rank of a given score? a. The percent of cases of a distribution below and above a given score. b. The percent of cases of a distribution below the given score. c. The percent of cases of a distribution above the given score. d. The percent of cases of a distribution within the given score.