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Central Tendency

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CENTRAL TENDENCY

Mean, Median, and Mode

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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

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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

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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

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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

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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

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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

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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:

0)( XX

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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

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Central Tendency

Characteristics of The Mean Changing a score Introducing a new score or

removing a score Adding or subtracting a constant

from each score Multiplying or dividing a constant

from each score

10

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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

The weighted mean

M =ΣX

N

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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:

M =Σ f.Xc

N

X Xc f f.Xc

20 - 24

15 - 19

10 - 14

5 - 9

0 - 4

22

17

12

7

2

1

4

7

5

3

22

68

84

35

6

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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

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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

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X0 1 2 3 4 5

5

4

3

2

1

f

X0 1 2 3 4 5

5

4

3

2

1

f

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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

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How to count the median?

Mdn = XLRL +

0.5N – f BELOW

LRL f TIED

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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

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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

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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

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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

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WHEN TO USE THE MEDIAN1. 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

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WHEN TO USE THE MEDIAN2. Undetermined values

Occasionally, we will encounter a situation in which an individual has an unknown or undetermined score

Person Time (min.)

123456

811121317

Never finished

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

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WHEN TO USE THE MEDIAN2. 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

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WHEN TO USE THE MEDIAN3. 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)

5 or more43210

322364

Notice that is impossible to compute a mean for these data because you cannot find ΣX

f

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WHEN TO USE THE MEDIAN4. Ordinal scale

when score are measured on an ordinal scale, the median is always appropriate and is usually the preferred measure of central tendency

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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

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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

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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

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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

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POSITIVELY SKEWED DISTRIBUTION

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NEGATIVELY SKEWED DISTRIBUTION