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Explain concept and the use of central tedency?

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10/24/2007

A central tendency is a number that expresses something "central" about a sample of values (which could be test scores, temperatures, etc...). Measures of central tendency include the mean, the median, and the mode. The Mean is equal to the average of all the values. Thus, the Mean is equal to the sum of all the values (add them all up) divided by the total number of values in your set or sample. This average tells you nothing about what your highest and lowest values are (the range). However, ... The Median is equal to the the number which, if you were to arrange your values from lowest to highest, falls exactly in the middle of your distribution of values. So, if you have 41 values, for instance, the Median would be the 21st value, and there would be 20 values equal to or smaller than the Median, and 20 values equal to or larger than the median. If, on the other hand, there were 100 values, the median would be the average of the 50th and 51st values in the distribution. The median tells you nothing, however, about what values occur "most often" in your distribution. So.... There is the mode, which is equal to the value which occurs most often in your distribution. Simply count how many times each of your values occurs, and the mode= the one that occurs most often. The following is an example of a distribution which is highly "skewed" meaning that there are differences between the mean, median and mode for the set of values being observed. For discussion, let's say we are talking about test scores. The score values in the data set are: 1, 1, 1, 1, 1, 1, 1, 2 ,3,4, 4 , 5, 5 ,8 ,8 ,8, 8 ,9 ,9, 9, 9,10, 10, 10, 11, 11 ,12 The Number of values in the data set is 27, indicating that 27 people took the test, or that that one way or another, the test was taken 27 times by some number of people, thus we say, N= 27. The average or "mean" is the Sum of the the values (162) divided by N, so 162/27 = 6 ; Thus, the average value or "mean" is 6. Notice that 6 occurs not even once in this distribution. Thus, even though the average is 6, your chances of obtaining a 6 (for instance if these were test scores) would only be 1 in 27! So, let's try looking at the Median- how many people's test scores will be above and below the median? We divide N= 27 by 2. 27/2=13.5, so it is the 14th value that equals the median. Counting from either end, the 14th value is 8. So, you can assume that if you score an 8 on the test, you are scoring at the 50th percentile, yet still "above the average." But how likely are you to get an 8? Well, It turns out that the most common score to get on this test is "1" which makes the Mode=1. That tells you there is a higher liklihood of getting a 1 than any other score. None-the-less, you should not necessarily expect to get a "1" on the test, since 20 out of 27 scores will be higher than that, and 14 out of the 27 scores will be "above average." The distribution of scores has a shape which can be approximated by drawing a line over the top of these number towers. 1

1

1

1 8 9

1 8 9 10

1 4 5 8 9 10 11

1 2 3 4 5 8 9 10 11 12

This distribution of numbers is said to be skewed since it looks like a declining slope, rather than a more or less evenly distributed set that would look more like a "hill." An inclining slope would also be considered a skewed distribution. Looking a these numbers, you could safely if you don't score "1" on the test, you will probably score somewhere between 8 and 10, and are very likely to score between 4 and 11. This distribution could represent, just as an example, a test where you get one point for writing your name, and then you either know most of the answers, or you know almost none of them, and get a coupld right by guessing. That is assuming there were only 12 possible points on the test, which may not be the case. It could very well be a distribution of scores on a 20 item test, which would then say something else about the test. Lets say that the test involves 20 possible points, one point per question, and that we were expecting that scores would be "normally distributed." That means we were expecting a mean of 10, an average of 10, and a mode of 10, with 86% of the scores falling within one standard deviation of the mean (that is another discussion), but basically "in the middle of" the distribution with 8 % on each end of the curve where the highest and lowest scores are represented, creating a curve that looks like a hill. If, despite, our expectation, we got scores as listed above, it would tell us that our assumption may not be true for the sample population we are studying, and that there are some factors impacting the way people score on the test that we need to look at more closely.