Measures of Central Tendency

Section 2.3

StatisticsMrs. SpitzFall 2008

Larson/Farber Ch 2

Objectives:

§ How to find the mean, median and mode of a population and a sample

§ How to find a weighted mean and the mean of a frequency distribution

§ How to describe the shape of a distribution as symmetric, uniform or skewed.

Assignment: pp. 62-66 #1-42

Larson/Farber Ch 2

Measure of central tendency

§ Is a value that represents a typical, or central entry of a data set. The three most commonly used measures of central tendency are mean, median, and the mode.

Larson/Farber Ch 2

Symbol Description

§ It would be a good idea now to start looking at the symbols which will be part of your study of statistics.

The uppercase Greek letter sigma; indicates a summation of values

X A variable that represents quantitative data

N Number of entries in a population

The lowercase Greek letter mu; the population mean

x Read as “x bar;” the sample mean

Larson/Farber Ch 2

Measures of Central Tendency

Mean: The sum of all data values divided by the number of values

For a population: For a sample:

Median: The point at which an equal number of values fall above and fall below

Mode: The value with the highest frequency

Larson/Farber Ch 2

0 2 2 2 3 4 4 6 40

2 4 2 0 40 2 4 3 6

Calculate the mean, the median, and the mode

Mean:

Median: Sort data in order

The middle value is 3, so the median is 3.

Mode: The mode is 2 since it occurs the most times.

An instructor recorded the average number of absences for his students in one semester. For a random sample the data are:

Larson/Farber Ch 2

Median: Sort data in order.

Mode: The mode is 2 since it occurs the most times.

The middle values are 2 and 3, so the median is 2.5.

0 2 2 2 3 4 4 6

Calculate the mean, the median, and the mode.

Mean:

2 4 2 0 2 4 3 6

Suppose the student with 40 absences is dropped from the course. Calculate the mean, median and mode of the remaining values. Compare the effect of the change to each type of average.

Larson/Farber Ch 2

Weighted Mean and Mean of Grouped Data

§ Sometimes data sets contain entries that have a greater effect on the mean than do other entries. To find the mean of such data sets, you must find the weighted mean.

w

wxx

)(

The weighted mean is the mean of a data set whose entries have varying weights. A weighted mean is given by the equation to the left where w is the weight of each entry x.

Larson/Farber Ch 2

Mean of Grouped Data

§ The mean of a frequency distribution for a sample is approximated by:

Where x and f are the midpoints and frequencies of a class, respectively.

n

fxx

)(

Larson/Farber Ch 2

Finding the mean of a frequency distribution

1. Find the midpoint of each class.

2. Find the sum of the products of the midpoints and the frequencies.

3. Find the sum of the frequencies

4. Find the mean of the frequency distribution.

Larson/Farber Ch 2

UniformSymmetric

Skewed right Skewed left

Mean = Median

Mean > Median Mean < Median

Shapes of Distributions