Measures of Central Tendency (MCT) 1. Describe how MCT describe data 2. Explain mean, median & mode 3. Explain sample means 4. Explain “deviations around mean”

2 More Statistical Notation An important symbol is ∑, it is the Greek letter ∑ called sigma This symbol means to sum (add) You will see it used in notations such as ∑ X. This is pronounced as the “sum of X” and means to find the sum of the X scores

Why Is It Important to Know about MCT?

4 Central Tendency MCT answer the question: –“Are the scores generally high scores or generally low scores?” What are they? –A MCT is a score that summarizes the location of a distribution on a variable –It is the score that indicates where the center of the distribution tends to be located

5 The Mode The most frequently occurring score is called the mode The mode is typically used to describe central tendency when the scores reflect a nominal scale of measurement

6 Unimodal Distributions When a polygon has one hump (such as on the normal curve) the distribution is called unimodal.

7 Bimodal Distributions When a distribution has two scores that are tied for the most frequently occurring score, it is called bimodal.

The Median

9 The median (Mdn) is the score at the 50th percentile The median is used to summarize ordinal or highly skewed interval or ratio scores

10 Determining the Median When data are normally distributed, the median is the same score as the mode. When data are not normally distributed, follow the following procedure: –Arrange the scores from lowest to highest. –If there are an odd number of scores, the median is the score in the middle position. –If there are an even number of scores, the median is the average of the two scores in the middle.

The Mean

12 The Mean The mean is the score located at the exact mathematical center of a distribution The mean is used to summarize interval or ratio data in situations when the distribution is symmetrical and unimodal

13 Determining the Mean The formula for the sample mean is

14 Sample Mean versus Population Mean is the sample mean. It is a sample statistic. is the sample mean. It is a sample statistic. The mean of a population is a parameter. It is symbolized by m (pronounced “mew”). is used to estimate the corresponding population mean m. is used to estimate the corresponding population mean m.

15 On a perfect normal distribution all three measures of central tendency are located at the same score. Central Tendency and Normal Distributions

16 Central Tendency and Skewed Distributions

Deviations Around the Mean

18 Deviations A score’s deviation is equal to the score minus the mean. In symbols, this is The sum of the deviations around the mean always equals 0.

19 More About Deviations When using the mean to predict scores, a deviation indicates our error in prediction. A deviation score indicates a raw score’s location and frequency relative to the rest of the distribution.