1. CENTRAL TENDENCY
Mean, Median, and Mode

2. OVERVIEW
The general purpose of descriptive statistical methods is to organize and summarize a set score
Perhaps the most common method for summarizing and describing a distribution is to find a single value that defines the average score and can serve as a representative for the entire distribution
In statistics, the concept of an average or representative score is called central tendency

3. OVERVIEW
Central tendency has purpose to provide a single summary figure that best describe the central location of an entire distribution of observation
It also help simplify comparison of two or more groups tested under different conditions
There are three most commonly used in education and the behavioral sciences: mode, median, and arithmetic mean

4. The MODE
A common meaning of mode is ‘fashionable’, and it has much the same implication in statistics
In ungrouped distribution, the mode is the score that occurs with the greatest frequency
In grouped data, it is taken as the midpoint of the class interval that contains the greatest numbers of scores
The symbol for the mode is Mo

5. The MEDIAN
The median of a distribution is the point along the scale of possible scores below which 50% of the scores fall and is there another name for P50
Thus, the median is the value that divides the distribution into halves
It symbols is Mdn

6. The ARITHMETIC MEAN
The arithmetic mean is the sum of all the scores in the distribution divided by the total number of scores
Many people call this measure the average, but we will avoid this term because it is sometimes used indiscriminately for any measure of central tendency
For brevity, the arithmetic mean is usually called the mean

7. The ARITHMETIC MEAN
Some symbolism is needed to express the mean mathematically. We will use the capital letter X as a collective term to specify a particular set of score (be sure to use capital letters; lower-case letters are used in a different way)
We identify an individual score in the distribution by a subscript, such as X1 (the first score), X8 (the eighth score), and so forth
You remember that n stands for the number in a sample and N for the number in a population

8. Properties of the Mode
The mode is easy to obtain, but it is not very stable from sample to sample
Further, when quantitative data are grouped, the mode maybe strongly affected by the width and location of class interval
There may be more than one mode for a particular set of scores. In rectangular distribution the ultimate is reached: every score share the honor! For these reason, the mean or the median is often preferred with numerical data
However, the mode is the only measure that can be used for data that have the character of a nominal scale

9. Properties of the Median

10. Properties of the Mean
Unlike the other measures of central tendency, the mean is responsive to the exact position of reach score in the distribution
Inspect the basic formula ΣX/n. Increasing or decreasing the value of any score changes ΣX and thus also change the value of the mean
The mean may be thought of as the balance point of the distribution, to use a mechanical analogy. There is an algebraic way of stating that the mean is the balance point:

11. Properties of the Mean
The sums of negative deviation from the mean exactly equals the sum of the positive deviation
The mean is more sensitive to the presence (or absence) of scores at the extremes of the distribution than are the median or (ordinarily the mode
When a measure of central tendency should reflect the total of the scores, the mean is the best choice because it is the only measure based of this quantity

12. The MEAN of Ungrouped Data
The mean (M), commonly known as the arithmetic average, is compute by adding all the scores in the distribution and dividing by the number of scores or cases ΣX
M = N

13. The MEAN of Grouped Data
When data come to us grouped, or
when they are too lengthy for comfortable addition without the aid of a calculating machine, or
when we are going to group them for other purpose anyway,
we find it more convenient to apply another formula for the mean:
Σ f.Xc
M = N
X Xc f f.Xc
5 - 9
0 - 4 22 17 12 7 2 1 4 7 5 3 22 68 84 35 6

14. The MEDIAN of Ungrouped Data
Method 1: When N is an odd number
 list the score in order (lowest to highest), and the median is the middle score in the list
Method 2: When N is an even number
 list the score in order (lowest to highest), and then locate the median by finding the point halfway between the middle two scores

15. The MEDIAN of Ungrouped Data
Method 3: When there are several scores with the same value in the middle of the distribution
 1, 2, 2, 3, 4, 4, 4, 4, 4, 5
There are 10 scores (an even number), so you normally would use method 2 and average the middle pair to determine the median
By this method, the median would be 4

16. f f 5 4 3 2 1 5 4 3 2 1 X X

17. The MEDIAN of Grouped Data
There are 10 scores (an even number), so you normally would use method 2 and average the middle pair to determine the median. By this method the median would be 4
In many ways, this is a perfectly legitimate value for the median. However when you look closely at the distribution of scores, you probably get the clear impression that X = 4 is not in the middle
The problem comes from the tendency to interpret the score of 4 as meaning exactly 4.00 instead of meaning an interval from 3.5 to 4.5

18. How to count the median?
0.5N – f BELOW LRL
Mdn = XLRL +
f TIED

19. THE MODE
The word MODE means the most common observation among a group of scores
In a frequency distribution, the mode is the score or category that has the greatest frequency

20. SELECTING A MEASURE OF CENTRAL TENDENCY
How do you decide which measure of central tendency to use? The answer depends on several factors
Note that the mean is usually the preferred measure of central tendency, because the mean uses every score score in the distribution, it typically produces a good representative value
The goal of central tendency is to find the single value that best represent the entire distribution

21. SELECTING A MEASURE OF CENTRAL TENDENCY
Besides being a good representative, the mean has the added advantage of being closely related to variance and standard deviation, the most common measures of variability
This relationship makes the mean a valuable measure for purposes of inferential statistics
For these reasons, and others, the mean generally is considered to be the best of the three measure of central tendency

22. SELECTING A MEASURE OF CENTRAL TENDENCY
But there are specific situations in which it is impossible to compute a mean or in which the mean is not particularly representative
It is in these condition that the mode an the median are used

23. WHEN TO USE THE MEDIAN Extreme scores or skewed distribution
When a distribution has a (few) extreme score(s), score(s) that are very different in value from most of the others, then the mean may not be a good representative of the majority of the distribution.
The problem comes from the fact that one or two extreme values can have a large influence and cause the mean displaced

24. WHEN TO USE THE MEDIAN Undetermined values
Occasionally, we will encounter a situation in which an individual has an unknown or undetermined score
Person Time (min.)
Notice that person 6 never complete the puzzle. After one hour, this person still showed no sign of solving the puzzle, so the experimenter stop him or her 1 2 3 4 5 6 8 11 12 13 17
Never finished

25. WHEN TO USE THE MEDIAN Undetermined values
There are two important point to be noted:
The experimenter should not throw out this individual’s score. The whole purpose to use a sample is to gain a picture of population, and this individual tells us about that part of the population cannot solve this puzzle
This person should not be given a score of X = 60 minutes. Even though the experimenter stopped the individual after 1 hour, the person did not finish the puzzle. The score that is recorded is the amount of time needed to finish. For this individual, we do not know how long this is

26. WHEN TO USE THE MEDIAN
Open-ended distribution
A distribution is said to be open-ended when there is no upper limit (or lower limit) for one of the categories
Number of children (X) f
5 or more 4 3 2 1 3 2 6 4
Notice that is impossible to compute a mean for these data because you cannot find ΣX

27. WHEN TO USE THE MEDIAN Ordinal scale
when score are measured on an ordinal scale, the median is always appropriate and is usually the preferred measure of central tendency

28. WHEN TO USE THE MODE Nominal scales
Because nominal scales do not measure quantity, it is impossible to compute a mean or a median for data from a nominal scale
Discrete variables  indivisible categories
Describes shape
the mode identifies the location of the peak (s). If you are told a set of exam score has a mean of 72 and a mode of 80, you should have a better picture of the distribution than would be available from mean alone

29. CENTRAL TENDENCY AND THE SHAPE OF THE DISTRIBUTION
Because the mean, the median, and the mode are all trying to measure the same thing (central tendency), it is reasonable to expect that these three values should be related
There are situations in which all three measures will have exactly the same or different value
The relationship among the mean, median, and mode are determined by the shape of the distribution

30. SYMMETRICAL DISTRIBUTION SHAPE
For a symmetrical distribution, the right-hand side will be a mirror image of the left-hand side
By definition, the mean and the median will be exactly at the center because exactly half of the area in the graph will be on either side of the center
Thus, for any symmetrical distribution, the mean and the median will be the same

31. SYMMETRICAL DISTRIBUTION SHAPE
If a symmetrical distribution has only one mode, it will also be exactly in the center of the distribution. All three measures of central tendency will have same value
A bimodal distribution will have the mean and the median together in the center with the modes on each side
A rectangular distribution has no mode because all X values occur with the same frequency. Still the mean and the median will be in the center and equivalent in value

32. SYMMETRICAL DISTRIBUTION SHAPE

33. POSITIVELY SKEWED DISTRIBUTION

34. NEGATIVELY SKEWED DISTRIBUTION