STA220H5 Chapter Notes - Chapter 2.3: Variance, Squared Deviations From The Mean

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26 Apr 2018
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2.3-2.4
The mean of a set of quantitative data is the sum of the measurements, divided by
the number of measurements contained in the data set.
Formula for a sample mean
= 

[Note: 
 =(x1 + x2 + . . . + xn). for more details and examples on this summation
notation, see appendix A.]
Symbols for the sample mean and the population mean
in this text, we adopt a general policy of using Greek letters to represent numerical
descriptive measures of the population and roman letters to represent
corresponding descriptive measures of the sample. The symbols for the mean are
= Sample mean μ = population mean
The median of a quantitative data set is the middle number when the measurements
are arranged in ascending (or descending) order.
Calculating a sample median M
Arrange the n measurements from the smallest to the largest.
1. If n is odd, M is the middle number.
2. If n is even, M is the mean of the middle two numbers.
Symbols for the sample and population median
M = sample median
η = population median
A data set is said to be skewed if one tail of the distribution has more extreme
observations than the other tail.
Detecting skewness by comparing the mean and the median
If the data set is skewed to the right, then typically the median is less than the mean.
If the data set is symmetric, then the mean equals the median.
If the data set is skewed to the left, then typically the mean is less than the median.
The mode is the measurement that occurs most frequently in the data set.
The range of a quantitative data set is equal to the largest measurement minus the
smallest measurement.
The sample variance for a sample of n measurements is equal to the sum of the
squared deviations from the mean, divided by (n-1). The symbol s2 is used to
represent the sample variance.
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Document Summary

The mean of a set of quantitative data is the sum of the measurements, divided by the number of measurements contained in the data set. + xn). for more details and examples on this summation notation, see appendix a. ] = sample mean = population mean. The median of a quantitative data set is the middle number when the measurements are arranged in ascending (or descending) order. Arrange the n measurements from the smallest to the largest: if n is odd, m is the middle number, if n is even, m is the mean of the middle two numbers. A data set is said to be skewed if one tail of the distribution has more extreme observations than the other tail. Detecting skewness by comparing the mean and the median. If the data set is skewed to the right, then typically the median is less than the mean. If the data set is symmetric, then the mean equals the median.

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