measures of central tendency statistics.pptx

lectures

1. Measures of Central Tendency and Dispersion/Variability

2. INTRODUCTION A measure of central tendency is a single value that attempts to describe a set of data (like scores) by identifying the central position within that set of data or scores. As such measures of central tendency are sometimes called measures of central location. Central tendency refers to the center of a distribution of observations.

3. INTRODUCTION Where do scores tend to congregate? In a test of 100 items, where are most of the scores? Do they tend to group around the mean score of 50 or 80? Perhaps you are most familiar with the mean (often called the average ).

4. INTRODUCTION Where do scores tend to congregate? In a test of 100 items, where are most of the scores? Do they tend to group around the mean score of 50 or 80? Perhaps you are most familiar with the mean (often called the average ). But, there are two other measures of central tendency namely the median and the mode

5. INTRODUCTION If measures of central tendency indicate where scores congregate, the measures of variability indicate how spread out a group of scores is or how varied the scores are or how far they are from the mean? Common measures of dispersion are range , interquartile range , variance and standard deviation .

6. THE MEASURES OF CENTRAL TENDENCY

7. The mean, mode and median are valid measures of central tendency but under different conditions, one measure becomes more appropriate than the others. For example, if the scores are extremely high or extremely low, the median is a better measure of central tendency since mean is affected by extremely high and extremely low scores.

8. THE MEAN The mean (or average or arithmetic mean) is the most popular and well known measure of central tendency. The mean is equal to the sum of all values in the data set divided by the number of values in the data set.

9. THE MEAN Mean = Sum of all the values/ total number of values

10. THE MEAN 70 72 75 77 78 80 84 87 90 92 EXAMPLE 1. Find the mean score of the ten students in Graduate School class who got the following scores in a 100 – item test: The mean score of the group of 10 students is the sum of all their scores divided by 10

11. THE MEAN 70 72 75 77 78 80 84 87 90 92 = 805/10 = 80.5

12. THE MEAN 70 72 75 77 78 80 84 87 90 92 Mean= 80.5 There are six scores below the average score of the group (70, 72, 75, 77, 78, 80) and there are 4 scores above the mean of the group (84, 87, 90, 92)

13. THE MEAN 29 30 21 15 30 18 29 EXAMPLE 2. Find the mean score of the 7 close friends of Anne in their PE exam which is composed of 30 items, the scores are the following : The mean score of the group of friends of Anne is the sum of all their scores divided by 7

14. THE MEAN 29 30 21 15 30 18 29 = 172/7 = 24.57

15. THE MEAN 29 30 21 15 30 18 29 Mean= 24.57 There are 3 scores below the mean of the group (15, 18, 21) and there are 4 scores above the mean (29, 29, 30,30).

16. Why not Use the Mean? The mean has one main disadvantage. It is susceptible to the influence of outliers. These are unusual compared to the rest of the data set by being small or large in numerical value. For example, consider the scores of 10 Grade 12 students in a 100-item Statistics test below: 5 38 56 60 67 70 73 78 79 95

17. Why not Use the Mean? 5 38 56 60 67 70 73 78 79 95 The mean score for these ten Grade 12 students is 62.1. However, inspecting the raw data suggests that this mean score may not be the best way to accurately reflect the score of the typical Grade 12 students, as most students have scores in the 5 to 95 range. The mean is being skewed by the extremely low and extremely high scores.

18. THE MEDIAN The median is the middle score for a set of scores arranged from lowest to highest. The mean is less affected by extremely low and extremely high.

19. THE MEDIAN How do we find the median? 65 55 89 56 35 14 56 55 87 45 92 To determine the median, first we have to rearrange the scores into order of magnitude (from smallest to largest) 14 35 45 55 55 56 56 65 87 89 92 O D D

20. THE MEDIAN Our median is the score at the middle of the distribution. In this case, 56 is the median. There are 5 scores before it and 5 scores after it. 14 35 45 55 55 56 56 65 87 89 92 O D D

21. THE MEDIAN How do we find the median? 65 55 89 56 35 14 56 55 87 45 To determine the median, first we have to rearrange the scores into order of magnitude (from smallest to largest) 14 35 45 55 55 56 56 65 87 89 E VE N

22. THE MEDIAN 14 35 45 55 55 56 56 65 87 89 E VE N Take the middle two scores (55 and 56) and compute the average of two scores. Median = 55+56 2 = 55.5 This gives us a more reliable picture of the tendency of the scores. There are indeed scores of 55 and 56 in the score distribution

23. THE MODE The mode is the most frequent score in our data set. On a histogram or bar chart it represents the highest bar If a score of the number of times an option is chosen in a multiple choice of test you can therefore consider the mode as being the most popular option.

24. THE MODE Determine the mode in the score distribution below: 65 55 89 56 35 14 56 55 87 45 There are two most frequent scores 55 and 56 . So we have a score distribution with two modes, hence a bimodal distribution.

25. THE MODE Determine the mode in the score distribution below: 20 17 25 10 20 10 17 11 20 15 16 20 11 There is only one most frequent score which is 20. The mode is 20, an unimodal distribution.

26. SEATWORK #1! 75 85 95 95 70 60 45 Find the mean, median and mode of the following data below:

27. SEATWORK #2! The scores of 15 students in a Math test, 50 items test are the following: 21 23 13 42 50 36 27 28 39 30 41 32 34 45 10 Calculate the mean of the scores. Find the median and modal score.

28. Normal and Skewed Distribution

29. A score distribution a sample has a “normal distribution” when most of the values are aggregated around the mean, and the number of values decrease as you move below or above the mean Normal distribution will look like a bell-shaped curve. Normal distribution

30. If the mean is equal to the median and median is equal to the mode, the score distribution shows a perfectly normal distribution.

31. If mean is less than the median and the mode, the score distribution is a negatively skewed distribution, the scores tend to congregate at the upper end of the score distribution.

32. If the mean is greater than the median and the mode , the score distribution is a positively skewed distribution, tend to congregate at the lower end of the score distribution.

33. If scores tend to be high because teacher taught very well and students are highly motivated to learn, the score distribution tends to be negatively skewed i.e. the scores will tend to be high. On the other hand, when teacher does not teach well and students are poorly motivated, the score distribution tends to be positively skewed which means that scores tend to below.

34. Outcome-based Teaching-Learning and Score Distribution If teachers teach in accordance with the principles of outcome-based teaching-learning and so align content and assessment with the intended learning outcomes and re-teach till mastery what has/have not been understood as revealed in the formative assessment process then student scores in the assessment phase of the lesson will tend to congregate on the higher end of the score distribution Score distribution will be a negatively skewed.

35. Outcome-based Teaching-Learning and Score Distribution On the other hand, if what teachers teach and assess are not aligned with the intended learning outcomes, the opposite will be true. Score distribution will be positively skewed which means that the scores tend to congregate on the lower end of the score distribution.