COMPARISON MEASURES OF CENTRAL TENDENCY & VARIABILITY EXERCISE 8/5/2013. MEASURE OF CENTRAL TENDENCY: MODE (Mo) MEASURE OF CENTRAL TENDENCY: MODE (Mo)

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1 COMPARISON MEASURES OF CENTRAL TENDENCY & VARIABILITY Prepared by: Jess Roel Q. Pesole CENTRAL TENDENCY -what is average or typical in a distribution Commonly Measures: 1. Mode. Median 3. Mean quantified characteristics of a distribution VARIABILITY -extent to which scores are spread out in a distribution Measures: 1. Range. Standard 3. Variance (Mo) The most frequent score in the distribution For ungroupeddata, it is the score or category that occurs most often in a distribution Example (for ungrouped data): 1,, 3, 4, 4, 5, 5, 6, 6, 6, 7, 8, 9, 10 Mo = 6 (Mo) For grouped data, it is the midpoint of the interval containing the largest number of cases Example: Mo = (44 +40) / = 4 Therefore, Mo = 4. Class Interval f (Mo) MAJOR MINOR Find the mode of the following series of numbers. 1. 4, 1, 0, 19, 18, 17, 16, 15, 14, 1., 0, 19, 18, 18, 18, 18, 16, 16, , 7, 7, 6, 6, 6, 6, 4, 4, 1

2 Determine the mode for the following frequency distribution. Class Interval f cf c% N = 50 MEDIAN (Mdn) Scale value below which 50 percent of scores fall Similar to P 50 For ungroupeddata, it is the centermost score if the number of scores are odd If it is even, it is the average of the two centermost scores Position of the median = (N + 1) / MEDIAN (Mdn) Example (for ungrouped scores): 11, 1, 13, 16, 17, 0, 5 N = 7 (odd) Therefore, the median is 16 11, 1, 13, 16, 17, 0, 5, 6 N = 8 Mdn= () = 16.5 Therefore, the median is 16.5 Position of Median = (N+ 1) / = (7 + 1) / = 8 / = 4 Position of Median = (N+ 1) / = (8 + 1) / = 9 / = 4.5 MEDIAN (Mdn) For grouped data, you can use the same formula as P 50 Example: Mdn= + = (150)(.) 63 = (0.1667)(75 63) = (0.1667)(1) = = Therefore, the median is Class Interval f cf C% N =150 Find the median of the following series of numbers. 1. 4, 1, 0, 19, 18, 17, 16, 15, 14, 1., 0, 19, 18, 18, 18, 18, 16, 16, , 7, 7, 6, 6, 6, 6, 4, 4, Determine the median for the following frequency distribution. Class Interval f cf c% N = 50

3 Sum of the scores divided by the number of scores = = Where: = mean of a sample = mean of a = summation = raw scores N= number of scores Example: The following are scores from a sample of 10 math scores. Solve for the mean of the scores. 11, 1, 13, 14, 16, 17, 17, 0, 5, 6 = = = =17.1 CONCLUSION: The mean from the sample of math scores is If you are given grouped data: = = Where: = mean of a sample = mean of a = summation = frequency of the interval m= midpoint of the interval N= number of scores OVERALL The weighted mean, the mean of means The sum of the mean of each group times the number of scores in the group, divided by the sum of the number of scores in each group = Where: = number of cases in a particular group = mean of a particular group = total number of cases in all groups combined Example: The following are the means of final grades from three blocks in psychology, as well as the number of students in each block. Determine the overall mean of the final grades for the three blocks. Section 1: 1 = 85; N 1 = 95 Section : = 7; N = 5 Section 3: 3 = 79; N 3 = 18 = = (18)(79) = = 11, = =81.86 CONCLUSION: The mean final grade for the three blocks is The following are productivity scores for four departments in a company, as well as its corresponding number of workers. Compute for the overall mean productivity for all four departments: Section 1: N 1 = 0; 1 = 10 Section : N = 15; = 14 Section 3:N 3 = 18; 3 = 15 Section 4:N 4 = ; 4 = 8 3

4 1 = 4 11 Deviation distance and direction of any raw score from the mean Deviation = X - Example: 9, 8, 6, 5, Mean = 6 X DEVIATION MEASURES OF CENTRAL TENDENCY AND SYMMETRY WHICH TO USE? -Used forany measurement scale, especiallythe nominal scale -Used when haste is necessary MEDIAN -Usedfor ordinalor interval data -Used when data is skewed, since it is not as sensitive to extreme scores compared to the mean - Used for interval or ratio data -Used when one has a symmetrical or normal distribution -Can also be useful in skewed distributions, since it is more flexible to advanced statistical analysis PROPERTIES OF THE 1. The mean is sensitive to the exact value of all the scores in the distribution.. The sum of the s about the mean equals zero. = (X i -) = 0 3. The mean is very sensitive to extreme scores. 4. The sum of the squared s of all the scores about their mean is a minimum. 5. Under most circumstances, of the measures used for central tendency, the mean is least subject to sampling variation. PROPERTIES OF THE MEDIAN 1. The median is less sensitive than the mean to extreme scores.. Under usual circumstances, the median is more subject to sampling variability than the mean but less subject to sampling variability than the mode. 4

5 RANGE Range-Difference between the highest and lowest scores in a distribution Range = Highest score lowest score Example: Range = 17 1 = 16 Measure of variability that reflects the typical from the mean = = = 1 = 1 SS= sum of squared s = s= sample ) The equation (Deviation = X -) was only limited to only two values Adding all s in a distribution would only lead to ZERO. In the past, the mean was used to determine the variability of a distribution. In this process, absolute s were used: = These days, the mean is no longer used since it is hard to use in advanced statistical analysis. To overcome the problems faced by a mean s, we use squared sinstead of absolute s From this we can get this formula, which is the equation for deviance: = However, the variance has the weakness of expressing variability in squared units (EXAMPLE: If you were asked to determine the variability of exam scores in a class, the variance will express it in terms of squared exam score.) As such, we get the square root of the deviance in order to reflect variability in the appropriate units; this becomes our sum of squares (SS) Raw Score Formula for Standard Deviation: = = 1 = s= sample ) X = sum of the squared raw scores N= total number of scores = mean square 5

6 Example: On a measure of authoritarianism (higher scores reflect greater tendency toward prejudice, ethnocentrism and submission to authority), seven students scored as follows: X X X = 37 X = 47 N = 7 Mean = Mean Square = PROPERTIES OF THE STANDARD DEVIATION 1. It gives a measure of dispersion relative to the mean.. It is sensitive to each score in the distribution. 3. It is stable with regards to sampling fluctuations. VARIANCE Square of the = = 1 SS= sum of squared s = s= sample ) VARIANCE Raw Score Formula for Variance: = = 1 = s= sample ) X = sum of the squared raw scores N= total number of scores = mean square RAWSCORES UNGROUPED GROUPED SUMMARY Scorewith the greatest frequency Scorewith the greatest frequency Midpointof the interval with the greatest frequency MEDIAN Middlemostscore, as determined by the position of the median Middlemost score, as determined by the position of the median Mdn (50 TH percentile)= + RAW SCORES SUMMARY UNGROUPED GROUPED = = = 6

7 SUMMARY PRACTICE RAW SCORES UNGROUPED GROUPED (sor ) VARIANCE (s or ) The scores of attitudes toward older people for 30 students were arranged in the following simple frequency distribution (higher scores indicate more favorable attitudes towards older people): Attitude Score Value f N = 30 Find the (a) mode, (b) median, and (c)mean. PRACTICE The scores on a religiosity scale (higher scores indicate greater commitment to religious expression) were obtained for 46 adults. For the following simple frequency distribution, calculate the three measures of central tendency. Score Value f cf N =46 7

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