Basic Statistics

Measures of Central Tendency

STRUCTURE OF STATISTICS

STATISTICS

DESCRIPTIVE

INFERENTIAL

TABULAR

GRAPHICAL

NUMERICAL

CONFIDENCEINTERVALS

TESTS OF HYPOTHESIS

1

Consider the following distribution of scores:

2

How do the red and blue distributions differ?

How do the red and green distributions differ?

Characteristics of Distributions

• Location or Center– Can be indexed by using a measure

of central tendency

• Variability or Spread– Can be indexed by using a measure

of variability

Consider the following distributions:

How do they differ?

Consider the following two distributions:

How do the green and red distributions differ?

Characteristics of Distributions

• Location or Central Tendency

• Variability

• Symmetry

• Kurtosis

STRUCTURE OF STATISTICS

STATISTICS

DESCRIPTIVE

INFERENTIAL

TABULAR

GRAPHICAL

NUMERICAL

CONFIDENCEINTERVALS

TESTS OF HYPOTHESIS

NUMERICAL

STRUCTURE OF STATISTICSNUMERICAL DESCRIPTIVE MEASURES

DESCRIPTIVE

TABULAR

GRAPHICAL

NUMERICAL

CENTRALTENDENCY

VARIABILITY

SYMMETRY

KURTOSIS

Measures of Central Tendency

Summarizing DataThe Mean

The Median

The Mode

Give you one score or measure that represents, or is typical of, an entire

group of scores

Give you one score or measure that represents, or is typical of, an entire

group of scores

frequency

score

Most scores tend to center toward a point in the distribution.

Central Tendency

35

41

73

84

35

47

56

35

52

39

35

84

69 7

7354

7 92

35

33

43

47

65

39

90

49

35

67

The Mean

The Median

The Mode

Frequency

Tables

Graphs

Measures of Central Tendency

49

52

41

84

52

41

Measurement scales

Frequency Tables & Graphs

Averaging

52

43

47

47

Tabulating

Graphing

Measures of Central Tendency

Are statistics that describe typical, average, or representative scores.

The most common measures of central tendency (mean,median, and mode) are quite different in conception and calculation.

These three statistics reflect different notions of the “center” of a distribution.

“The Mode”The score that occurs most frequently

In case of ungrouped frequency distribution

When observations have been grouped into classes, the midpoint of the class with the largest frequency is used as an estimate of the mode.

The mode of this distribution is estimated to be 52, the midpoint of the 51-53 class

In case of grouped frequency distribution

Unimodal Distribution -One Mode-

Bimodal Distribution –Two Modes-

Mode and Measurement Scales

1 2 1 3 3 2 3 3 3 1 2 1 2 3 3 2 1 2 3 2

1 2 3 4 4 3 4 3 2 4 4 2 1 2 4 4 3 2 3 4

112 132 112 113 112 150 125 114

68 56 39 56 44 56 45 56 75 81 67 59

Nationality

1=American

2=Asian

3=Mexican

Football Poll

1=first

2=second

3=third

4=fourth

IQ score Weight

Can you find a mode for each data?

3 4 112 56

Nominal Scale Ordinal Scale Interval Scale Ratio Scale

It is not affected by extremely large or small values and is therefore a valuable measure of central tendency when such values occur.

It can be found for ratio-level, interval-level, ordinal-level and nominal-level data

“The Mode”

The Median is the 50th percentile of a distribution

- The point where half of the observations fall below and half of the observations fall above

In any distribution there will always be an equal number of cases above and below the Median.

“The Median”

Location

Oh my !!Where is the

median?

For an odd number of untied scores (11, 13, 18, 19, 20)

11 12 13 14 15 16 17 18 19 20

The Median is the middle score when scores are arranged in rank order

Median Location = (N+1)/2 = 3rd

Median Score = 18

For an even number of untied scores (11, 15, 19, 20)

11 12 13 14 15 16 17 18 19 20

The Median is halfway between the two central values when scores are arranged in rank order

Md score=(15+19)/2=17

Median Location = (N+1)/2 = 2.5th

The Median of group of scores is that point on the number line such that sum of the distances of all scores to that point is smaller than the sum of the distances to any other point.

There is a unique median for each data set.

It is not affected by extremely large or small values and is therefore a valuable measure of central tendency when such values occur.

The Median can be computed for

•Ordinal-level data, or•Interval-level data, or•Ratio-level data.

Median and Levels of Measurement

1 2 1 3 3 2 3 3 3 1 2 1 2 3 3 2 1 2 3 2

1 2 3 4 4 3 4 3 2 4 4 2 1 2 4 4 3 2 3 4

112 132 112 113 112 150 125 114

68 56 39 56 44 56 45 56 75 81 67 59

Nationality Football Poll IQ score Weight

Can you find a median for each type of data?

No

Yes Yes Yes

The Mean

XN

XN

DefinitionDefinition:: For ungrouped data, the population mean is the sum of all the population values divided by the total number of population values. To compute the population mean, use the following formula.

Population mean

Population mean

Sigma

Population size

Population size

Individual valueIndividual value

Definition: Definition: For ungrouped data, the sample mean is the sum of all the sample values divided by the number of sample values. To compute the sample mean, use the following formula.

THE SAMPLE MEANTHE SAMPLE MEAN

X XnX-bar

Sigma

Individual value

Sample Size

Characteristics of The Mean

Center of Gravity of a Distribution

Center of Gravity of a Distribution

1 2 3 4 5 6 7 8

Mean

Data set

25 27

3129

35 37

33

31

31 31

31 31

31

31 31

How much error do you expect for each case?

The Mean6

-6

-2

4

-4

0 2

Deviation Scores

On average, I feel fine

It’s too cold!

It’s too hot!

The Mean of group of scores is the point on the number line such that sum of the squared differences between the scores and the mean is smaller than the sum of the squared difference to any other point. If you summed the differences without squaring them, the result would be zero.

Mean and Measurement Scales

Every set of interval-level and ratio-level data has a mean.

1 2 3

Nationality

1=American

2=Asian

3=Mexican

Nominal data

1 2 3 1 2 3 1 2 3

IQ Test

Ordinal data Interval data Ratio data

Football Poll1=first2=second3=third

Weight

2 2 2 2NO YES YESNO

All the values are included in computing the mean.

X Xn

A set of data has a unique mean and the mean is affected by unusually

large or small data values.

3 5 7 9

The Mean

1

5

1

654

9

55.5

3

5

• Every set of interval-level and ratio-level data has a mean.

• All the values are included in computing the mean.

• A set of data has a unique mean.• The mean is affected by unusually large or

small data values.• The arithmetic mean is the only measure of

central tendency where the sum of the deviations of each value from the mean is zero.

The Relationships between Measures of Central Tendency

and Shape of a Distribution

Symmetric Unimodal

Normal Distribution

Mean=Median=Mode

Positively Skewed Distribution

Mode < Median < Mean

Mode

Median

Mean

The median falls closer to the mean than to the mode

With unimodal curves of moderate asymmetry, the distance from the median to the mode is approximately twice that of the distance between the median and

the mean

Negatively Skewed Distribution

Mode > Median > Mean

Mode

Median

Mean

The median falls closer to the mean than to the mode

Bimodal Distribution

Mean=Median

Mode Mode

Mode1 < Mean=Median < Mode2

If two averages of a moderately skewed frequency distribution are known, the third can be approximated.

The formulae are:

Mode = Mean - 3(Mean - Median)

Mean = [3(Median) - Mode]/ 2

Median = [2(Mean) + Mode]/ 3

Measures of Central Tendency as Inferential Statistics

Parameters

Statistics

Sampling

Mean Median Mode

Mean Median Mode

Difference Between Parameter and Statistics

Sampling Errors

As inferential measures, the Mean will be used much more frequently than the Median or Mode.

Why ?

On the average, there is less sampling error associated with the Mean than with the Median, and the Mode tends to have more sampling error than the Median. In other words, the difference between the statistic X and the Mean tends to be less than for the corresponding values for the sample Median (Md) and population median (Mdpop).

SUMMARY

There are three common measures of central tendency. The mean is the most widely used and the most precise for inferential purposes and is the foundation for statistical concepts that will be introduced in subsequent class. The mean is the ratio of the sum of the observations to the number of observations. The value of the men is influenced by the value of every score in a distribution. Consequently, in skewed distributions it is drawn toward the elongated tail more than is the median or mode.

The median is the 50th percentile of a distribution. It is the point in a distribution from which the sum of the absolute differences of all scores are at a minimum. In perfectly symmetrical distributions the median and mean have same value. When the mean and median differ greatly, the median is usually the most meaningful measure of central tendency for descriptive purposes.

The mode, unlike the mean and median, has descriptive meaning even with nominal scales of measurement. The mode is the most frequently occurring observation. When the median or mean is applicable, the mode is the least useful measure of central tendency. In symmetrical unimodal distribution the mode, median, and mean have the same value.