Measure of Central Tendency Vernon E. Reyes

A single number that repreresents the average Useful way to describe a group Central tendency – it is generally located towards the middle or center of the distribution where most of the data tend to be concentrated Well-known measures of central tendency are: mode, the median, and the mean

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Mode The mode (Mo) is the only measure of central tendency used for NOMINAL DATA like religion, college major It can also describe any level of measurement The Mo is found through INSPECTION rather than COMPUTATION

Example 2 3 1 1 6 5 4 1 4 4 3 What id the Mo? = ____ 2 3 1 1 6 5 4 1 4 4 3 What id the Mo? = ____ Note: the Mo is NOT the frequency (f = 4) (Mo = 1)

The mode can be unimodal or bimodal

Median When ordinal or interval data are arranged in order or size, its possible to locate the median (Md or Mdn) – the middlemost point in a distribution The position of the median value can be located by inspection or by formula Position of the median = N+1 / 2

Odd or Even For odd number of cases (N) the median is easy to find 11 12 13 16 17 20 25 Using the formula (7+1) / 2 = 4 Therefore the fourth place is the median which is equal to 16

Odd or Even For even number of cases (N) the median is always the point above or below where 50% of the cases will fall. 11 12 13 16 ! 17 20 25 26 Using the formula (8+1) / 2 = 4.5 Therefore the fourth place is the median which is equal to 16.5

Other note! If the data are not in order from low to high (or high to low), you should put them in order first before trying to locate the median!

The MEAN The arithmetic mean X = mean (read as x bar X = raw score N = Total number of score Σ = sum (greek capital letter sigma)

Mo vs Md vs Mean “center of gravity” A number that is computed which balances the scores above and below it To understand the meaning of the MEAN we must look at the deviation DEVIATION = X – X Where X = any raw score X = mean of the distribution

example +5 - 5 ------------------------------ X X – X 9 +3 8 +2 6 0 5 -1 2 -4 X = 6 Notice that if we add all the deviations it will always equal to zero! (+)5 + (-)5 = 0 Later we shall discuss standard deviation +5 - 5

Another example ------------------------------ X X – X 1 2 3 5 6 7 X = ? Find the mean! Find the deviations!

The Weighted Mean The mean of means! Example: Section 1: X 1 = 85 N 1 = 28 Section 2: X 2 = 72 N 2 = 28 Section 3: X 3 = 79 N 2 = 28 85 + 72 + 79 = 236 3 3 = 78.97

Formula for weighted mean with unequal sizes Xw = Σ Ngroup Xgroup Ntotal Xw = N1X1 + N2X2 +N3X3

The Weighted Mean The mean of means! Example: unequal N Section 1: X 1 = 85 N 1 = 95 Section 2: X 2 = 72 N 2 = 25 Section 3: X 3 = 79 N 2 = 18 95(85) + 25(72) + 18(79) = 8075+1800+1422 138 138 11,297 138 = 81.86

Lets compare Mean, Median and Mode! X f How old did you get married? 31 1 23 2 30 22 29 21 28 20 3 27 19 4 26 18 25 Find the mean median and mode 24