Basic Statistics Measures of Central Tendency

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

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.

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? They differ in symmetry.

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.

Characteristics of Distributions Location or Central Tendency Variability Symmetry Kurtosis

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

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

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

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

Frequency Tables & Graphs Measures of Central Tendency 73 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

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. In case of grouped frequency distribution The mode of this distribution is estimated to be 52, the midpoint of the 51-53 class

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

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 1 2 1 3 3 2 3 3 3 1 2 1 2 3 3 2 1 2 3 2 4 1 2 3 4 4 3 4 3 2 4 4 2 1 2 4 4 3 2 3 4 112 56 68 56 39 56 44 56 45 56 75 81 67 59 112 132 112 113 112 150 125 114 Nationality 1=American 2=Asian 3=Mexican Football Poll 1=first 2=second 3=third 4=fourth IQ score Weight

“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

“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

Median Location = (N+1)/2 = 3rd 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 Median Location = (N+1)/2 = 2.5th Md score=(15+19)/2=17

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.

No Yes Yes Yes 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 No Yes Yes Yes Nationality Football Poll IQ score Weight Can you find a median for each type of data?

The Mean

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

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

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

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

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

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.

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

All the values are included in computing the mean.

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

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

Normal Distribution Symmetric Unimodal 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 Mode1 < Mean=Median < Mode2 Mode Mode

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 Mean Median Mode Difference Between Parameter and Statistics Sampling Sampling Errors Statistics Mean Median Mode

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.