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Chapter 13 Statistics

WHAT YOU WILL LEARN • Sampling techniques • Misuses of statistics • Frequency distributions • Histograms, frequency polygons, stem-and-leaf displays • Mode, median, mean, and midrange • Percentiles and quartiles

WHAT YOU WILL LEARN • Range and standard deviation • z-scores and the normal distribution • Correlation and regression

Linear Correlation and Regression Section 8 Linear Correlation and Regression

Linear Correlation Linear correlation is used to determine whether there is a relationship between two quantities and, if so, how strong the relationship is.

Linear Correlation The linear correlation coefficient, r, is a unitless measure that describes the strength of the linear relationship between two variables. If the value is positive, as one variable increases, the other increases. If the value is negative, as one variable increases, the other decreases. The variable, r, will always be a value between –1 and 1 inclusive.

Scatter Diagrams A visual aid used with correlation is the scatter diagram, a plot of points (bivariate data). The independent variable, x, generally is a quantity that can be controlled. The dependent variable, y, is the other variable. The value of r is a measure of how far a set of points varies from a straight line. The greater the spread, the weaker the correlation and the closer the r value is to 0. The smaller the spread, the stronger the correlation and the closer the r value is to 1 or -1.

Correlation

Correlation

Linear Correlation Coefficient The formula to calculate the correlation coefficient (r) is as follows:

Level of Significance What is the minimum value or r needed to assume that a correlation exists between the variables? Level of Significance, denoted by , is used to identify the cutoff between results attributed to chance and results attributed to an actual relationship between the two variables. Critical Values, or cutoff scores, are sometimes used for determining whether two variables are related.

Example: Words Per Minute versus Mistakes There are five applicants applying for a job as a medical transcriptionist. The following shows the results of the applicants when asked to type a chart. Determine the correlation coefficient between the words per minute typed and the number of mistakes. Determine whether a correlation exists at =0.05 and 0.01. 9 34 Nancy 10 41 Kendra 12 53 Phillip 11 67 George 8 24 Ellen Mistakes Words per Minute Applicant

Solution We will call the words typed per minute x, and the mistakes y. List the values of x and y and calculate the necessary sums. WPM Mistakes x y x2 y2 xy 24 8 576 64 192 67 11 4489 121 737 53 12 2809 144 636 41 10 1681 100 410 34 9 1156 81 306 x = 219 y = 50 x2 =10,711 y2 = 510 xy = 2,281

Solution (continued) The n in the formula represents the number of pieces of data. Here n = 5.

Solution (continued)

Solution (continued) Since 0.86 is fairly close to 1, there is a fairly strong positive correlation. This result implies that the more words typed per minute, the more mistakes made.

Solution (continued)