Xi x skewness s n 1 n 2 3 skewness

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: kewness is positive (often above 1.0). -Mean will usually be more than the median. .35 Relative Frequency .30 .25 .20 .15 .10 .05 0 Skewness = 1.25 Distribution Shape: Skewness Example: Apartment Rents Seventy efficiency apartments were randomly sampled in a small college town. The monthly rent prices for these apartments are listed in ascending order as below. 4 25 440 450 465 480 510 575 430 440 450 470 485 515 575 430 440 450 470 490 525 580 435 445 450 472 490 525 590 435 445 450 475 490 525 600 435 445 460 475 500 535 600 435 445 460 475 500 549 600 435 445 460 480 500 550 600 440 450 465 480 500 570 615 440 450 465 480 510 570 615 Distribution Shape: Skewness Relative Frequency .35 .30 .25 .20 .15 .10 .05 0 Skewness = .92 Z-Scores The z-score is often called the standardized value. It denotes the number of standard deviations a data value xi is from the mean. xi x zi s Z-Scores An observation’s z-score is a measure of the relative location of the observation in a data set. A data value less than the sample mean will have a z-score less than zero. A data value greater than the sample mean will have a z-score greater than zero. A data value equal to the sample mean will have a z-score of zero. Z-Scores z-Score of Smallest Value (425) xi x 425 490.80 z 1.20 s 54.74 Standardized Values for Apartment Rents -1.20 -0.93 -0.75 -0.47 -0.20 0.35 1.54 -1.11 -0.93 -0.75 -0.38 -0.11 0.44 1.54 -1.11 -0.93 -0.75 -0.38 -0.01 0.62 1.63 -1.02 -0.84 -0.75 -0.34 -0.01 0.62 1.81 -1.02 -0.84 -0.75 -0.29 -0.01 0.62 1.99 -1.02 -0.84 -0.56 -0.29 0.17 0.81 1.99 -1.02 -0.84 -0.56 -0.29 0.17 1.06 1.99 -1.02 -0.84 -0.56 -0.20 0.17 1.08 1.99 -0.93 -0.75 -0.47 -0.20 0.17 1.45 2.27 -0.93 -0.75 -0.47 -0.20 0.35 1.45 2.27 Empirical Rule When the data are believed to approximate a bell-shaped distribution … The empirical rule can be used to determine the percentage of data values that must be within a specified number of standard deviations of the mean. The empirical rule is based on the normal distribution, which is covered in Chapter 6. Empirical Rule For data having a bell-shaped d68.26% of the values of a normal random variable istribution: are within +/- 1 standard deviation of its mean. 95.44% of the values of a normal random variable are within +/- 2 standard deviations of its mean. 99.72% of the values of a normal random variable are within +/- 3 standard deviations of its mean. Empirical Rule 99.72% 95.44% 68.26% m – 3 m – 1 m – 2 m m + 3 m + 1 m + 2 x Detecting Outliers An outlier is an unusually small or unusually large value in a data set. A data value with a z-score less than -3 or greater than +3 might be considered an outlier. It might be: -an incorrectly recorded data value -a data value that was incorrectly included in the data set -a correctly recorded data value that belongs in the data set Detecting Outliers The most extreme z-scores are -1.20 and 2.27 Using |z| > 3 as the criterion for an outlier, there are no outliers in this data set. Standardized Va...
View Full Document

Ask a homework question - tutors are online