1. MEASURES OF CENTRAL TENDENCY

2. Measures of Central Tendency
What is a measure of central tendency?
Measures of Central Tendency
Mode
Median
Mean
Shape of the Distribution
Considerations for Choosing an Appropriate Measure of Central Tendency

3. What is a measure of Central Tendency?
Numbers that describe what is average or typical of the distribution
You can think of this value as where the middle of a distribution lies.

4. WHY CAN’T THE MEAN TELL US EVRYTHING?
Mean describes Central Tendency, what the average outcome is. We also want to know something about how accurate the mean is when making predictions. The question becomes how good a representation of the distribution is the mean? How good is the mean as a description of central tendency -- or how good is the mean as a predictor? Answer -- it depends on the shape of the distribution. Is the distribution normal or skewed?

5. The Mode
The category or score with the largest frequency (or percentage) in the distribution.
The mode can be calculated for variables with levels of measurement that are: nominal, ordinal, or interval-ratio.

6. MODE AN EXAMPLE
Example: Number of Votes for Candidates for Mayor. The mode, in this case, gives you the “central” response of the voters: the most popular candidate. Candidate A – 11,769 votes The Mode: Candidate B – 39,443 votes “Candidate C” Candidate C – 78,331 votes

7. Measures of Central Tendency
One further parameter of a population that may give some indication of central tendency of the data is the mode
Define: mode = most frequently occurring value in the population
From the previous data we see:
65, 67, 68, 68, 69, 69, 71, 71, 71, 72, 72, 72, 73, 73, 73, 74, 74, 75, 75, 75, 75, 76, 76, 77, 77, 77, 77, 77, 77, 78, 78, 78, 78, 79, 79, 79, 79, 80, 81, 81, 81, 81, 81, 81, 81, 81, 82, 82, 83, 84, 85, 85, 85, 86, 86, 87, 87, 88, 89, 92
That the value 81 occurs 8 times mode = 81
Note! If two different values were to occur most frequently, the distribution would be bimodal. A distribution may be multi-modal.

8. MOST COMMON OUTCOME
Male Female

9. The score that divides the distribution into two equal parts, so that half the cases are above it and half below it. The median is the middle score, or average of middle scores in a distribution.

10. TO COMPUTE THE MEDIAN · then count number of observations = 10.
· first you rank order the values of X from low to high:  85, 94, 94, 96, 96, 96, 96, 97, 97, 98
· then count number of observations = 10.
· add 1 = 11.
· divide by 2 to get the middle score  the 5 ½ score
here 96 is the middle score score

11. EXAMPLE OF Median (N is odd)
Calculate the median for this hypothetical distribution:
TEMPERATURE Frequency Very High 2
High 3
Moderate 5
Low 7
Very Low 4
TOTAL

12. Median Exercise (N is even)
Calculate the median for this hypothetical distribution:
TEMPERATURE Frequency
Very High 5
High 7
Moderate
Low 7
Very Low 3
TOTAL 28

13. MEDIAN
Find the Median ,000

14. Measures of Central Tendency
A second measure of central tendency is the median
The median of a population of size N is found by
Arranging the individual measurements in ascending order, and
If N is odd, selecting the value in the middle of this list as the median (there will be the same number of values above and below the median)
If N is even find the values at position N/2 and N/2 + 1 in this list (call them xN/2 and xN/2+1) and let median be given by the formula median = (xN/2 + xN/2+1)/2 or be the value halfway between these two measurements.
Note! When N is even the median will usually not be an actual value in the population

15. Measures of Central Tendency
We now find the median of the population of temperature readings
87, 85, 79, 75, 81, 88, 92, 86, 77, 72, 75, 77, 81, 80, 77,
73, 69, 71, 76, 79, 83, 81, 78, 75, 68, 67, 71, 73, 78, 75,
84, 81, 79, 82, 87, 89, 85, 81, 79, 77, 81, 78, 74, 76, 82,
85, 86, 81, 72, 69, 65, 71, 73, 78, 81, 77, 74, 77, 72, 68
Arrange these 60 measurements in ascending order
65, 67, 68, 68, 69, 69, 71, 71, 71, 72, 72, 72, 73, 73, 73, 74, 74, 75, 75, 75, 75, 76, 76, 77, 77, 77, 77, 77, 77, 78, 78, 78, 78, 79, 79, 79, 79, 80, 81, 81, 81, 81, 81, 81, 81, 81, 82, 82, 83, 84, 85, 85, 85, 86, 86, 87, 87, 88, 89, 92
Since N/2 = 30 and both the 30th and 31st values in the list are the same, we obtain median = 78

16. The Mean
The arithmetic average obtained by adding up all the scores and dividing by the total number of scores.

17. Formula for the Mean
“X bar” represents MEAN

18. X = (Σ X) / N If X = {3, 5, 10, 4, 3} X = (3 + 5 + 10 + 4 + 3) / 5
FINDING THE MEAN
X = (Σ X) / N
If X = {3, 5, 10, 4, 3}
X = ( ) / 5
= / 5 =

19. Calculating the mean with grouped scores
where: fx = a score multiplied by its frequency

20. From the table we obtain
Mean: Grouped Scores
From the table we obtain
Class Class Midpoint (x) Total (f) Frequency f*x
69.5 –
74.5 –
79.5 –
84.5 –
89.5 –

21. Merits AND Demerits of MEAN
1.It is easy to calculate
2.It is easy to follow
DEMERITS
1.It is highly effected by extreme values
2.It cannot average the ratios and percentage properly

22. MEAN FOR DISCRETE SERIES
Number of People(x) Frequency(f) TOTAL 581

23. Shape of the Distribution
Symmetrical (mean is about equal to median)
Negatively (example: years of education)
mean < median
Positively (example: income)
mean > median
Bimodal (two distinct modes)
Multi-modal (more than 2 distinct modes)
Draw Examples on the board

24. Distribution Shape

25. Measures of Central Tendency
Next we show where each of these parameters occur in the frequency distribution graph for this tabulated data.
Frequency % 42 39 36 33 30 27 24 21 18 15 12 9 6 3
Mean =
Median = 78 x
median
Midrange = 78.5
Mode = 81
mean x x x x x
Temperature

26. THANK YOU