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# Day 3 descriptive statistics

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This slides introduce the descriptive statistics and its differences with inferential statistics. It also discusses about organizing data and graphing data.

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### Day 3 descriptive statistics

1. 1. Thursday, October 9, 2014 1
2. 2. Refers to methods and techniques used for describing, organizing, analyzing, and interpreting numerical data.
3. 3.  The field of statistics is often divided into two broad categories : descriptive statistics and inferential statistics.  Descriptive statistics transform a set of numbers or observations into indices that describe or characterize the data.
4. 4.  Thus, descriptive statistics are used to classify, organize, and summarize numerical data about a particular group of observations.  There is no attempt to generalize these statistics, which describe only one group, to other samples or population.
5. 5.  In other words, descriptive statistics are used to summarize, organize, and reduce large numbers of observations.  Descriptive statistics portray and focus on what is with respect to the sample data, for example: 1. What is the average reading grade level of the fifth graders in the school?” 2. How many teachers found in-service valuable?” 3. What percentage of students want to go to college?
6. 6. Inferential statistics (sampling statistics), involve selecting a sample from a defined population and studying that sample in order to draw conclusions and make inferences about the population.
7. 7. 100,000 fifth-grade students take an English achievement test 100,000 fifth-grade students take an English achievement test Researcher randomly samples 1,000 students scores Researcher randomly samples 1,000 students scores Used to describe the sample Used to describe the sample Based on descriptive statistics to estimate scores of the entire population of 100,000 students Based on descriptive statistics to estimate scores of the entire population of 100,000 students
8. 8. Focuses on ways to organize numerical data and present them visually with the use of graphs. One way to organize your data is to create a frequency distribution.  Various software programs, such as Excel, can easily produce graphs for you.
9. 9. Allows researchers and educators to describe, summarize, and report their data. By organizing data, they can compare distributions and observe patterns.
10. 10. In most cases, the original data we collect is not ordered or summarized.  Therefore, after collecting data, we may want to create a frequency distribution by ordering and tallying the scores.
11. 11. A seventh-grade social studies teacher wants to assign end-of term letter grades to the twenty-five students in her class. After administering a thirty-item final examination, the teacher records the students’ test scores.
12. 12. 27 25 30 24 19 16 28 24 17 21 23 26 29 23 18 22 20 17 24 23 21 22 28 26 25 These scores show the number of correct answers obtained by each students on the social studies final examination. Next, the researcher can create a frequency distribution by ordering and tallying these scores.
13. 13. Score Frequency Score Frequency 30 29 28 27 26 25 24 23 11212233 22 21 20 19 18 17 16 2211121
14. 14.  The researcher/teacher may want to group every scores together into class interval to assign letter grade to the students. Class interval (5 points) Mid point Frequency 26-30 21-25 16-20 28 23 18 7 12 6 Σ 25
15. 15. A researcher of experimental research administered a thirty-item reading comprehension test. Next, the researcher records the students’ reading scores. Please, create a frequency distribution of thirty scores with class intervals of five points and interval midpoints. 74 80 66 69 63 65 61 62 58 59 57 58 57 57 55 56 53 54 51 52 49 50 47 48 31 44 43 36 39 41
16. 16. Graphs are usually to communicate information by transforming numerical data into a visual form. Graphs allow us to see relationships not easily apparent by looking at the numerical data. There are various forms of graphs, each are appropriate for a different type of data.
17. 17. In drawing histogram and frequency polygon, the vertical axis always represents frequencies, and the horizontal axis always represents scores or Class interval (Mid point). The lower values of both vertical and horizontal axes are recorded at the intersection of the axes (at the bottom left side).
18. 18. Lowest Highest Highest Lowest
19. 19. Frequency distribution in the following table can be depicted using two types of graphs, a histogram or a frequency polygon. Score Frequency 654321 124321
20. 20. A Frequency Distribution of Twenty-five Scores with class Intervals and Midpoints Class Interval Midpoint Frequency 38-42 33-37 28-32 23-27 18-22 13-17 8-12 3-7 40 35 30 25 20 15 10 5 13465321
21. 21. The following data are unorganized examination score of two groups taught with different method Group A (Language laboratorium) N=30 Group A (Language laboratorium) N=30 Group B (Non-language laboratorium) N=30 Group B (Non-language laboratorium) N=30 15 12 11 18 15 15 9 19 14 13 11 12 18 15 14 16 17 15 17 13 14 13 15 17 19 17 18 16 11 16 14 18 689 14 12 12 10 15 12 9 16 17 12 87 15 5 14 13 13 12 11 13 11
22. 22. The following data are unorganized examination score of two groups taught with different method a. Arrange the frequency distribution of scores! b. Arrange interval frequency distribution of scores of five points! c. Figure the histogram of the scores! d. Figure the frequency Polygon of the scores! e. Take a conclusion from the histogram and frequency polygon you graph.
23. 23. They are descriptive statistics that measure the central location or value of sets of scores. A measure of central tendency is a summary score that is used to represent a distribution of scores. It is a summary score that represents a set of scores. They are used widely to summarize and simplify large quantities of data.
24. 24. The mode of the distribution is the score that occurs with the greatest frequency in that distribution. Score Frequency Mode 12 11 10 98765 11234211 We can see that the score of 8 is repeated the most (four times); therefore, the mode of the distribution is 8.
25. 25. The mode of the distribution is the score that occurs with the greatest frequency in that distribution. Score Frequency Mode 12 11 10 98765 11234211 We can see that the score of 8 is repeated the most (four times); therefore, the mode of the distribution is 8.
26. 26. The mode in the distribution below is? Score Score 16 22 17 22 18 22 18 23 20 We can see that the score of 22 is repeated the most (three times); therefore, the mode of the distribution is 22.
27. 27.  The median is the middle point of a distribution of scores that are ordered  Fifty percent of the scores are above the median , and 50 percent are below it. Score Median 10 876421 The score 6 is the median because there are three scores above it and three below it.
28. 28.  If the distribution has an even number of scores, the median is the average of the two middle scores. Score 20 16 12 10 Median Two middle scores 877642 Thus, the median in the score above is (7+8):2= 7.5
29. 29.  It is the “arithmetic average” of a set of scores.  It is obtained by adding up the scores and dividing that sum by the number of scores.  The statistical symbol for the mean of a sample is χ (pronounced “ex bar”).  A raw score is represented in statistics by the letter X.  A raw score is score as it was obtained on a test or any other measure, without converting it to any other scale.
30. 30.  The statistical symbol for the population mean is μ, the Greek letter mu (pronounced “moo” or “mew”).  The statistical symbol for “sum of” is Σ (the capital Greek letter sigma).  The formula for calculating the mean is or
31. 31.  The statistical symbol for the population mean is μ, the Greek letter mu (pronounced “moo” or “mew”).  The statistical symbol for “sum of” is Σ (the capital Greek letter sigma).
32. 32. Calculation of Mean if we have obtained the sample of eight scores : 17,14,14,13.10,8,7,7 Answer: By using raw score Score Score 17 10 14 14 13 877 Σ X= 17+14+14+13+10+8+7+7=90 N=8 Thus, the mean is
33. 33. Calculation of Mean if we have obtained the sample of eight scores : 17,14,14,13.10,8,7,7 Answer: By score distribution Scor e Frequenc y F x Score 17 14 13 10 87 121112 17 28 13 10 8 14 8 90 Σ X= 17+28+13+10+8+14=90 N=8 Thus, the mean is
34. 34. Are used to show the differences among the scores in a distribution. We use the term variability or dispersion because the statistics provide an indication of how different, or dispersed, the scores are from one another.
35. 35. The range is the simplest; but also least useful, measure of variability. It is defined as the distance between the smallest and the largest scores. It is calculated by simply subtracting the bottom, or lowest, score from the top, or highest score. Range = XH- XL XH = the highest score XL = the lowest score
36. 36. Determine the range and the mean from the following sets of figures : a. 1,4,9,11,15,19,24,29,34 b. 14,15,15,16,16,16,18,18,18 Answer a: Mean= ........ Range ........... Answer b: Mean= ........ Range .........
37. 37. The distance between each score in a distribution and the mean of that distribution is called the deviation score. The mean of the deviation scores is called the standard deviation (SD) The standard deviation tells you” how close the scores are to the mean.”
38. 38. The SD describes the mean distance of the scores around the distribution mean. Squaring the SD give us another index of variability, called the VARIANCE. The Variance is needed in order to calculate the SD (Standard Deviation).
39. 39. If the standard deviation is a small numbers, this tells you that the scores are “bunched together” close to the mean.  If the standard deviation is a large number, this tells you that the scores are “spread out” a greater distance from the mean.
40. 40. The formula for standard deviation is: for group scores
41. 41. The variance (S2) is a measure of dispersion that indicates the degree to which scores cluster around the mean. Computationally, the variance is the sum of the squared deviation scores about the mean divided by the total number of scores/the total number of scores minus one. or
42. 42. or  If we have only five scores. It is very likely that such a small group of scores is a sample, rather than a population. Therefore, we computed the variance and SD for these scores, treating them as a sample, and used a denominator of N-1 in the computation. When, on the other hand, we consider a set of scores to be a population, we should use a denominator of N to compute the variance.
43. 43. For any distribution of scores, the variance can be determined by following five steps: Step 1:calculate the mean: (ΣX/N) Step 2: calculate the deviation scores: Step 3: Square each deviation score : Step 4: Sum all the deviation scores: Step 5 : Divide the sum by N:
44. 44. Calculate the standard deviation from the following scores: 2,3,3,4,5,5,5,6,6,8 Answer: Calculate the variance by using 5 steps Step 1:calculate the mean: (ΣX/N) Step 2: calculate the deviation scores: Step 3: Square each deviation score : Step 4: Sum all the deviation scores: Step 5 : Divide the sum by N:
45. 45. Raw Scores 2334555668 2-4.7=-2.7 3-4.7=-1.7 3-4.7=-1.7 4-4.7=-0.7 5-4.7=0.3 5-4.7=0.3 5-4.7=0.3 6-4.7=1.3 6-4.7=1.3 8-4.7=3.3 7.29 2.89 2.89 0.49 0.09 0.09 0.09 1.69 1.69 10.89 28.10 28.10/10 = 2.81 Thus the Standard Deviation is
46. 46. Calculate the standard deviation from the following scores: 20,15,15,14,14,14,12,10,8,8 Answer: Calculate the variance by using 5 steps Step 1:calculate the mean: (ΣX/N) Step 2: calculate the deviation scores: Step 3: Square each deviation score : Step 4: Sum all the deviation scores: Step 5 : Divide the sum by N: