Measure of central tendency
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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

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A single number that repreresents the average
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




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
Example

  • 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)



Median
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
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 even1
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
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 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
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


Example1

------------------------------

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

example

+5

- 5


Another example

------------------------------

X X – X

-----------------------------

1

2

3

5

6

7 X = ?

Find the mean!

Find the deviations!

Another example


The weighted mean
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
Formula for weighted mean with unequal sizes

Xw = Σ Ngroup Xgroup

Ntotal

Xw = N1X1 + N2X2 +N3X3

Ntotal


The weighted mean1
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



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