Measures of Central Tendency - Grouped Data

Measures of Central tendency

1. Measures of Central Tendency:

2. MEASURES of CENTRAL TENDENCY 2 Measures of central tendency provides a very convenient way of describing a set of scores with a single number that describes the PERFORMANCE of the group. It also defined as a single value that is used to describe the “ center ” of the data. There are three commonly used measures of central tendency, these are the following: - Mean - Median - Mode

3. MEASURES of CENTRAL TENDENCY 3 Mean, median, and mode are single representative values that can be used to described what is typical in a large set of data. In statistics, these values collectively referred to as measures of central tendency .

4. M ean The mean is the most commonly used measure of average or central tendency. Denoted by the symbol x̅ , the mean can be determined by getting the sum of the set of data and dividing it by the number of data.   4

5. 5 Computation of mean for: Ungrouped data Grouped data x̅   x̅  

6. (Ungrouped Data) Find the mean in the following scores. 6 x (scores) 25 14 18 13 20 14 18 12 17 12 15 10 15 10 15 x̅ = 228 15 = 15.2   UNGROUPED DATA – data which is also known as raw data is data that is not sorted into categories, classified as, or otherwise grouped. A set of data basically a list of numbers.

7. where gives the individual scores and the corresponding weights of each score.   7 Example: Find the grade point average of Jen if she obtained the grades of 87, 90, 82, and 80 each having 2, 1.8, 1.5, and 1.2 units respectively. Sometimes each individual score has different levels of significance, or weights, and you are asked to get the mean. In this case, you have to specifically obtain the weighted mean, which can be determined by dividing the sum of the products and the weights by the total weights. Thus,        

8. MEAN for GROUPED DATA 8 Grouped Data are the data or scores that are arranged in a frequency distribution. Frequency Distribution is the arrangement of scores according to category of classes including the frequency.

9. TERMINOLOGIES (Recall) Class Interval Class Limits Class Size(Class Width) Frequency Class Boundaries Classmark (Midpoint)

10. CLASS INTERVAL Class intervals or classes, are the groups of values that function as row classifiers. In the above example, the class intervals are 1-5, 6-10, 11-15, 16-20, etc.

11. CLASS LIMITS Class limits are the smallest and largest possible values that can fall into each class interval. The smaller number in the class interval is called the lower limit, whereas the larger number is the upper limit. The numbers 1, 6, 11, …31 are lower limits, and the numbers 5, 10, 15, 20, … 35 are upper limits.

12. CLASS SIZE or CLASS WIDTH Class size or class width is the number of values that are contained in each class. The size is uniform to all classes of the frequency distribution table. Class size can be determined by subtracting two successive lower limits or two consecutive upper limits. In the previous sample frequency distribution table, the class size is 5 since the difference between two consecutive lower limits or upper limits is 5 (i.e., 6-1=5 or10-5=5).

13. FREQUENCY Frequency is the number of cases falling into each class interval. The previous table is just an example of simple frequency distribution. A complete distribution table includes other parts such as class boundary, class mark, relative frequency, and cumulative frequency.

14. CLASS BOUNDARIES Class boundaries, also known as true limits, are obtained by subtracting 0.5 to the lower limit and adding 0.5 to upper limit of each class. This is always true when the class limits are whole numbers. However, note that if data are in one decimal place, you subtract or add 0.05 to get the class boundaries, and 0.005 for two-decimal limits. Class boundaries are very useful when dealing with continuous type of data.

15. CLA Class mark, denoted by , is the midpoint of a class interval. This can be obtained by dividing the sum of the upper limit (UL) and the lower limit (LL) of the class interval by 2 SS MARK   Class mark, denoted by , is the midpoint of a class interval. This can be obtained by dividing the sum of the upper limit (UL) and the lower limit (LL) of the class interval by 2, thus, . For example in the first class the class mark will be  

16. Score Class Boundary Class Mark Frequency 1-5 0.5-5.5 3 3 6-10 5.5-10.5 8 2 11-15 10.5-15.5 13 11 16-20 15.5-20.5 18 1 21-25 20.5-25.5 23 20 26-30 25.5-30.5 28 8 31-35 30.5-35.5 33 5 Class size = 5 N = 50

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18. To simplify the process, follow the steps: Compute the class marks of each class. Multiply each class mark by the corresponding frequencies. Get the summation of all products in step 2. Divide the summation of the products by the total number of observations.

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20. MEDIAN The median refers to the middle value in a set of data. It may be obtained by arranging the scores in either ascending or descending order and then locating the   22

21. EXAMPLE Determine the median of the following scores: 9,12, 10, 16, 8, 11, and 10. Solution : The scores when arranged in ascending order will give 8, 9, 10, 10,11, 12, and 16. Since there are 7 scores, the position of the midpoint will be: which is 10.   23

22. EXAMPLE Identify the median of the following grades: 82, 75, 88, 90, 77, 78, 97, 93, 91, and 95. Solution : Arrange the scores in ascending order. 75, 76, 78, 82, 88, 90, 91, 93, 95, 97 Since there are 7 scores, the position of the midpoint will be:    

23. Median for Grouped Data The concepts of class boundary and less than cumulative frequency are crucial in determining the median for grouped data. To compute for the mean of grouped data, use the formula     Where L = lower class boundary of the median class N = total frequency F = cumulative frequency before the median class = frequency of the median class i = class size/width Note : The formula above only works when the class interval are arranged in ascending order.  

24. Here are the steps in getting the median for grouped data: Compute the less than cumulative frequencies. Locate the median class. Since the median is the middle value in the distribution, it can be found on the first class interval whose less than cumulative frequency is at least . Substitute the appropriate values in the formula, then solve.  

25. MODE The value that appears most frequently in a given data is called the mode. A distribution may be classified according to its number of modes. For example, a set of data is called unimodal when it has exactly one mode, bimodal when it contains two modes, and multimodal when it has many modes. In some instances, a set of data may have no mode. 31

26. F ind the mean: 38 Age Frequency(f) x fx 10-12 5 11 55 13-15 8 14 112 16-18 5 17 85 19-21 10 20 200 22-24 2 23 46 N=30 fx =498 Age Frequency(f) x fx 10-12 5 11 55 13-15 8 14 112 16-18 5 17 85 19-21 10 20 200 22-24 2 23 46 N=30 x̅ = 498 30 = 16.60  

27. F ind the median: 39 Age Frequency(f) L.C.B < cf 10-12 5 9.5 5 13-15 8 12.5 13 16-18 5 15.5 18 19-21 10 18.5 28 22-24 2 21.5 30 N=30 x̃ =   x̃ =15.5   x̃ =15.5   x̃ =15.5   x̃ =15.5   x̃ =16.7  

28. F ind the mode: 40 Age Frequency(f) L.C.B 10-12 5 9.5 13-15 8 12.5 16-18 5 15.5 19-21 10 18.5 22-24 2 21.5 x=   =18.5   =18.5   =18.5   = 19.65        

29. Mean Median Mode Pros Most widely used measures of central tendency because of its high reliability Always a representative of the whole set of data Appropriate for interval and ratio scales of data Has only one value to describe a set of data Not easily affected by extreme values in the data Mostly appropriate when dealing with interval scales of data Has only one single value in a given set of data. Appropriate for nominal scales of data Easy to compute Cons Easily influenced by extremely high or extremely low scores or values in the distribution Not always a representative of the set of data Non-uniqueness of the value since there can be many modes in a set of data Not reliable Not always a representative Characteristics of Mean, Median, and Mode

30. Activity Read and analyze the given problem carefully. 42

31. problem ! 1. The table below shows the weights(kg) of members in a sport club. Calculate the mean, median and mode of the distribution. Masses 40-49 50-59 60-69 70-79 80-89 90-99 Frequency 6 8 12 14 7 3