1. Basic Statistics
Measures of Central Tendency

2. STRUCTURE OF STATISTICS
TABULAR
DESCRIPTIVE
GRAPHICAL
NUMERICAL
STATISTICS
CONFIDENCE INTERVALS
INFERENTIAL
TESTS OF HYPOTHESIS

3. Consider the following distribution of scores:
How do the red and blue distributions differ?
How do the red and green distributions differ?
Red blue differ by location. We can index that by indicating the centers and how different they are.
The red and green distributions differ by spread or variability of the data.

4. 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

5. Consider the following distributions:
How do they differ?
They differ in symmetry.

6. Consider the following two distributions:
How do the green and red distributions differ?
They differ in height or flatness or peakedness. This is called kurtosis.

7. Characteristics of Distributions
Location or Central Tendency
Variability
Symmetry
Kurtosis

8. STRUCTURE OF STATISTICS
TABULAR
DESCRIPTIVE
GRAPHICAL
NUMERICAL
NUMERICAL
STATISTICS
CONFIDENCE INTERVALS
INFERENTIAL
TESTS OF HYPOTHESIS

9. STRUCTURE OF STATISTICS NUMERICAL DESCRIPTIVE MEASURES
TABULAR
CENTRAL TENDENCY
DESCRIPTIVE
GRAPHICAL
NUMERICAL
VARIABILITY
SYMMETRY
KURTOSIS

10. Measures of Central Tendency
Summarizing Data
The Mean
The Median
The Mode
Give you one score or measure that represents, or is typical of, an entire group of scores

11. Most scores tend to center toward a point in the distribution.
frequency
score
Central Tendency

12. Frequency Tables & Graphs Measures of Central Tendency73 33 52
Averaging 67 43 35
The Mean
Frequency Tables
Tabulating 84 41 39 47 35 35 52
Graphing 84 49 35 47
The Median 90 35 52 47
Graphs 43 56 41 84 35 69
Measurement
scales 35 77
The Mode 39 47 65 92 52 41 49 47

13. 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.

14. “The Mode” The score that occurs most frequently
In case of ungrouped frequency distribution

15. When observations have been grouped into classes, the midpoint of the class with the largest frequency is used as an estimate of the mode.
In case of grouped frequency distribution
The mode of this distribution is estimated to be 52, the midpoint of the class

16. Unimodal Distribution -One Mode- Bimodal Distribution –Two Modes-

17. 3 4 112 56 Mode and Measurement Scales
Can you find a mode for each data?
Nominal Scale
Ordinal Scale
Interval Scale
Ratio Scale 3 4
112 56
Nationality
1=American
2=Asian
3=Mexican
Football Poll
1=first
2=second
3=third
4=fourth
IQ score
Weight

18. “The Mode”
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

19. “The Median” 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.
Oh my !!
Where is the median?
Location

20. Median Location = (N+1)/2 = 3rd
For an odd number of untied scores (11, 13, 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

21. For an even number of untied scores (11, 15, 19, 20)
The Median is halfway between the two central values when scores are arranged in rank order
Median Location = (N+1)/2 = 2.5th
Md score=(15+19)/2=17

22. 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.

23. The Median can be computed for
Ordinal-level data, or
Interval-level data, or
Ratio-level data.

24. No Yes Yes Yes Median and Levels of Measurement No
Yes
Yes
Yes
Nationality
Football Poll
IQ score
Weight
Can you find a median for each type of data?

25. The Mean

26. Definition: 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.
Sigma
Individual value
Population mean
Population size

27. THE SAMPLE MEAN
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.
Sigma
Individual value
X-bar
Sample Size

28. Characteristics of The Mean
Center of Gravity of a Distribution

29. Center of Gravity of a Distribution1 2 3 4 5 6 7 8
Mean

30. How much error do you expect for each case?
Deviation Scores -6 25 31 -4 27 31 31 31 31 2 -2 31 29 31 33
The Mean 6 4 35 31 31 37
Data set

31. On average,
I feel fine
It’s too hot!
It’s too cold!

32. 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.

33. 2 2 2 2 YES NO YES NO Mean and Measurement Scales 1 2 3 1 2 3 1 2 3
Every set of interval-level and ratio-level data has a mean.
Nominal data
Ordinal data
Interval data
Ratio data 2 2 2 2
YES NO
YES NO
Nationality
1=American
2=Asian
3=Mexican
Football Poll
1=first
2=second
3=third
IQ Test
Weight

34. All the values are included in computing the mean.

35. A set of data has a unique mean and the mean is affected by unusually large or small data values.1 1 3 3 5 7 9 9 5
5.5 5 6 5 5 4
The Mean

36. 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.

37. The Relationships between Measures of Central Tendency and Shape of a Distribution

38. Normal Distribution
Symmetric
Unimodal
Mean=Median=Mode

39. 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

40. Negatively Skewed Distribution
Mode
Median
Mean
Mode > Median > Mean
The median falls closer to the mean than to the mode

41. Bimodal Distribution Mode1 < Mean=Median < Mode2 Mode Mode

42. 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

43. Measures of Central Tendency as Inferential Statistics
Parameters
Mean Median Mode
Difference Between Parameter and Statistics
Sampling
Sampling Errors
Statistics
Mean Median Mode

44. 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).

45. 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.