University:
California State University, Northridge
Course:
MATH 140 | Introductory Statistics
Academic year:

2011
Views:

117

Pages:

29
Author:

Prince Sampson

Chapter 3 : Central Tendency O Overview i • Definition: Central tendency is a statistical measure to determine a single score that th t defines d fi th center the t off a distribution. – The goal of central tendency is to find the single score that is most typical or most representative of the entire group. – Measures of central tendency are also useful for making comparisons between groups of individuals or between sets of figures. • For example, example weather data indicate that for Seattle, Washington, the average yearly temperature is 53° and the average annual precipitation is 34 inches. O Overview i cont. t – • • • By comparison, the average temperature in Phoenix, Arizona, is 71 ° and the average precipitation is 7.4 inches. Clearly there are problems defining the Clearly, "center" of a distribution. Occasionally, you will find a nice, neat distribution like the one shown in Figure g 3.2(a), for which everyone will agree on the center. (See next slide.) But you should realize that other di ib i distributions are possible ibl and d that h there h may be different opinions concerning the definition of the center. (See in two slides) O Overview i cont. t • To deal with these problems, statisticians have developed three different methods for measuring central tendency: – Mean – Median M di – Mode O Overview i cont. t Negatively Skewed Distribution Bimodal Distribution Th M The Mean • • • • The mean for a population will be identified by the Greek letter mu, μ (pronounced "mew"), and the mean for a sample is identified by M or (read "xbar ). bar"). Definition: The mean for a distribution is the sum of the scores divided by the number of scores. The formula for the population mean is The formula for the sample mean uses symbols that signify sample values: Th M The Mean cont. t • In general, we will use Greek letters to identify characteristics of a population and letters of our own alphabet to stand for sample values. – Thus, Thus if a mean is identified with the symbol M, you should realize that we are dealing with a sample. – Also note that n is used as the symbol y for the number of scores in the sample. Th W The Weighted i ht d M Mean • Often it is necessary to combine two sets of scores and then find the overall mean for the combined group. • IIn summary, when h two t samples l are combined, the weighted mean is obtained as follows: Computing p g the Mean from a Frequency q y Distribution Table • The formula to calculate the mean from a frequency distribution table is as follows: ∑f X M = ────── n Ch Characteristics t i ti off the th Mean M • The mean has many characteristics that will be important in future discussions. – Changing a score • Changing the value of any score will ill change h the th mean. – Introducing a new score or removing a score • In general, general the mean is determined by two values: ∑X and N (or n). • Whenever either of these values are changed, g , the mean also is changed. Ch Characteristics t i ti off the th Mean M cont. t – Adding or subtracting a constant from each score • If a constant value is added to every score in a distribution, the same constant will be added to the mean. – Multiplying or dividing each score by a constant • If every score in a distribution is multiplied by (or divided by) a constant value, the mean will change h iin the h same way. Th M The Median di • • • • • Definition: The median is the score that divides a distribution in half so that 50% of the individuals in a distribution have scores at or below the median. Earlier when we introduced the mean Earlier, mean, specific symbols and notation were used to identify the mean and to differentiate a sample mean and a population mean. For the median, however, there are no symbols or notation. Instead, the median is simply identified b the by h word d median. di In addition, the definition and the computations for the median are identical for a sample and for a population. population Th M The Median di cont. t • • The goal of the median is to determine the midpoint of the distribution. This commonsense goal is demonstrated in the following two examples which show how the median for most distributions can be found simply by counting scores. Fig. 3-5, p. 83 Th M The Mode d • Definition: In a frequency distribution, the mode is the score or category g y that has the greatest frequency. – As with the median, there are no symbols or special notation used to id tif th identify the mode d or tto differentiate diff ti t between a sample mode and a population mode. – In addition, the definition of the mode is the same for a population and for a sample distribution. – Although a distribution will have only one mean and only one median, it is possible to have more than one mode. • Specifically, it is possible to have two or more scores that have the same highest frequency. Th M The Mode d cont. t • In a frequency distribution graph, the different modes will correspond to distinct, equally high peaks. • A distribution with two modes is said to be bimodal, and a distribution with more than two modes is called multimodal. Selecting g a Measure of Central Tendency • How do you decide which measure of central tendency to use? – The answer to this question depends on several factors. – Before B f we discuss di these th factors, f t however, note that you usually can compute two or even three measures of central tendency y for the same set of data. – Although the three measures often produce similar results, there are situations i i in i which hi h they h are very different (see Section 3.6). Wh tto use the When th M Median di • • Extreme scores or skewed distributions – When a distribution has a few extreme scores, scores that are very different in value from most of the others, then the mean may not be a good representative of the majority of the distribution. • The problem comes from the fact that one or two extreme values can have a large influence and cause the mean to be displaced. U d t Undetermined i d values l – Occasionally, you will encounter a situation in which an individual has an unknown or undetermined score. Wh tto use the When th M Median di cont. t • • Open-ended distributions – A distribution is said to be openended when there is no upper limit (or lower limit) for one of the categories. – The Th table t bl att the th upper right-hand i ht h d corner provides an example of an open-ended distribution, showing the number of p pizzas eaten during ga1 month period for a sample of n = 20 high school students. Ordinal scale – Many researchers believe that it is not appropriate to use the mean to describe central tendency for ordinal data. data Wh tto use the When th M Median di cont. t – When scores are measured on an ordinal scale, the median is always appropriate and is usually the preferred measure of central tendency. Wh tto use the When th M Mode d • • • Nominal scales – The primary advantage of the mode is that it can be used to measure and describe the central tendency for data that are measured on a nominal scale. Di Discrete t variables i bl – Recall that discrete variables are those that exist only in whole, indivisible categories. – Often, discrete variables are numerical values, such as the number of children in a family or the number of rooms in a house. Describing shape – Because the mode requires little or no calculation, it is often included as a supplementary measure along with the mean or median as a no-cost extra. extra Central Tendency y and the Shape p of the Distribution • • • • We have identified three different measures of central tendency, and often a researcher calculates all three for a single set of data. Because the mean mean, the median median, and the mode are all trying to measure the same thing (central tendency), it is reasonable to expect that these three values should be related. In fact, there are some consistent and predictable relationships among the three measures of central tendency. tendency Specifically, there are situations in which all three measures will have exactly the same value. Central Tendency y and the Shape p of the Distribution cont. • • • On the other hand, hand there are situations in which the three measures are guaranteed to be different. In part, the relationships among the mean, median, and mode are determined by the shape of the distribution. We will consider two general types of distributions. distributions – Symmetrical – Skewed Distributions S Symmetrical t i l Di Distributions t ib ti • • • • For a symmetrical distribution, the righthand side of the graph is a mirror image of the left-hand side. If a distribution is perfectly symmetrical, the median is exactly at the center because exactly half of the area in the graph will be on either side of the center. For a p perfectly y symmetrical y distribution with one mode, all three measures of central tendency, the mean, the median, and the mode, will have the same value. F a roughly For hl symmetrical i l distribution, di ib i the three measures are clustered together in the center of the distribution. S Symmetrical t i l Di Distributions t ib ti • • • On the other hand, a bimodal distribution that is symmetrical [see Figure 3.13(b)] will have the mean and median together in the center with the modes on each side. A rectangular distribution [see Figure 3.13(c)] has no mode because all X values occur with the same frequency. Still, the mean and the median are in the center of the distribution. Sk Skewed d Distributions Di t ib ti • • Distributions are not always symmetrical; quite often, they are lopsided, or skewed. – For example, Figure 3. 14(a) shows a positively skewed distribution. I skewed In k d di distributions, t ib ti especially i ll distributions for continuous variables, there is a strong tendency for the mean, median,, and mode to be located in predictable positions.

Chapter 3. Central Tendency

Related Documents

Recommended Documents

Description

Related Documents

Free up your schedule!

Take 5 seconds to unlock