Central Tendency The Aim • The main purpose of Measures of Central Tendency is to provide a single summary figure that best describes the central location of an entire distribution. • A measure of Central Tendency also helps to simplify comparison of two or more groups tested under different conditions Mode • Mode is the measure of most frequent score in the distribution. – The symbol for mode is Mo • For an ungrouped data, the score which has highest frequency is the mode • For a grouped data, the midpoint of the interval that contains the greatest number of the scores is the mode Median • Median is the 50th percentile in a distribution. That is the score below which half of the scores fall –The symbol for median is Mdn • To find the median, we need to rank order the scores. Median • For ungrouped/raw scores, median is the middle score of the distribution. – If n (or N) is odd, the median will be the score that has an equal number of score below and above it. • For the distribution of 0,7,9,10, and 15 the median is 9 – If n (or N) is even, there is no middle score. So, median is taken as the point halfway between the two score. • For the distribution of 2,4,5,6,8, and 10 the median is between 5 and 6. – Mdn = 5 + (6-5)/2 = 5.5 – If median fall between a repeating numbers, we will need to infer it by using interpolation. • For the distribution of 4,5,6,6,6,7,9, and 10, the median falls between second 6 and third 6. That is, median falls2/3 of the interval between 5.5 and 6.5 with interval width 1.0. • So, the median is 5.5 + (2/3)X 1.0= 5.5 + .67 = 6.17 Median • For grouped data, the median is 50th percentile. So, we can fallow the same steps that we used to calculate the percentile – Find which interval the score falls into. – Find how many cases are located in this interval – Find the difference between the percentile rank (median) and the cumulative frequency of the lower interval. That is, the number of the cases that we need to go up to reach the percentile rank – Calculate the proportion for the percentile and multiply it by the interval width – Add the result to the lower limit of the interval Arithmetic Mean • Arithmetic mean is the sum of score in a distribution divided by the total number of scores. – The symbol for mean is X bar for sample, and mu for population – We use the symbol “sum of …” in the formula • For raw scores, mean is equal to sum of X divided by n • For grouped scores, we multiply midpoint of each interval by f , sum these numbers, and divide it by n Prosperities of measures of Central Tendency • Mode: – It is not very stable from sample to sample. – There may be more than one mode for a particular set of scores – Mode is the only measure that can be used for the data collected by a nominal scale • So , Mode can be used for all kinds of scales Prosperities of measures of Central Tendency • Median: – Median is less sensitive to the presence of extreme scores • So, for strongly asymmetrical distributions or distributions with a few very deviant scores, the median may be the better choice for measuring central tendency – Median is a good choice for the scores that researcher cannot record the exact value for the upper end of distribution – Median is more stable than mode. It is more resistant to the influence of sampling fluctuation – Median cannot be used for nominal data. Prosperities of measures of Central Tendency • Mean: – Unlike the other measures of central tendency, the mean is responsive to the exact position of each score in distribution. – Mean is the balance point of a distribution. – The mean is more sensitive to the presence (or absence) of extreme scores – Mean cannot be used for nominal and ordinal data.