MATH125 Chapter 3 topics ANALYZING DATA NUMERICALLY

Notation  denotes the sum of a set of values. x is the variable usually used to represent the individual data values. n represents the number of values in a sample. N represents the number of values in a population.

Section 3.1 Measures of Central Tendency 1.mean 2.median 3.mode 3-3

Mean (or Average ) = (sum of all values)/number of observations Example: Data: Number of home runs hit by Babe Ruth as a Yankee 54, 59, 35, 41, 46, 25, 47, 60, 54, 46, 49, 46, 41, 34, 22 The mean number of home runs hit in a year is: The MEAN

Notations for the Mean pronounced ‘mu’ and denotes the mean of all values in a population. x = n  x x pronounced ‘x-bar’ and denotes the mean of a set of sample values. N µ =  x x

If x 1, x 2, …, x N are the N observations of a variable from a population, then the population mean, µ, is 3-6

If x 1, x 2, …, x n are the n observations of a variable from a sample, then the sample mean,, is 3-7

The median is the value that lies in the middle of the data when arranged in ascending order. We use M to represent the median. Example 1: Ordered list of home run hits by Babe Ruth: n=15 Median = 46 8 th Example 2: Ordered list of home run hits by Roger Maris in 1961: n=10 Median = (23+26)/2=24.5 The MEDIAN

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What does it mean for a statistic to be resistant ? 3-10

Mean versus Median The mean is sensitive to extreme values. The mean is a good representation of the central tendency when the data values don’t contain extremes (that is data values that are very small or very large compared to the majority of the data). When there are extreme values, then the median is a better representation of the central tendency.

A quantity is said to be resistant if extreme values (very large or small) relative to the data do not affect its value substantially. The Median is resistant to extreme values. The Mean is not. 3-12

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EXAMPLE Describing the Shape of the Distribution The following data represent the asking price of homes for sale in Lincoln, NE. Source: 79,995128,950149,900189,900 99,899130,950151,350203, ,200131,800154,900217, ,000132,300159,900260, ,000134,950163,300284, ,700135,500165,000299, ,950138,500174,850309, ,900147,500180,000349, Find the mean and median. 2.Use the mean and median to identify the shape of the distribution. 3.Verify your result by drawing a histogram of the data.

Answers 1)The mean is $168,320 and the median is $148,700. 2)The distribution is skewed right. 3)Histogram: 3-15

The MODE  The Mode is the value that occurs most frequently.  The Mode is not always unique.  A data set may be: Bimodal Multimodal No Mode

EXAMPLE Finding the Mode of a Data Set The data on the next slide represent the Vice Presidents of the United States and their State of birth. Find the mode for the variable State. 3-17

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The mode is New York.

Tally data to determine most frequent observation 3-21 Summary

Exercise: Finding the mean for a larger data set. SUV price data: $14,655 $14,799 $15,605 $16,395 $16,798 $17,990 $19,300 $20,000 $21,995 $22,195 $22,708 $23,240 $23,405 $23,920 $25,176 $25,999 $26,185 $26,268 $27,815 $27,910 $28,680 $28,950 $29,099 $29,249 $30,585 $30,645 $31,985 $32,250 $32,950 $33,595 $33,790 $34,590 $35,550 $36,300 $38,175 $41,188 $42,660 $54,950 $56,000 $63,500 Find the Mean and Median of these prices. Answers: Too tedious to do by adding all then dividing by number of observations. Solution 1: Use the STAT feature on your TI-83 calculator. Solution 2: Use Excel built-in functions: o median(data) o average (data)

Using the calculator to find statistics on data: Step by step: Press the "STAT" button. With "EDIT" highlighted select "1:Edit" by pressing "ENTER". If there is data in List 1 (L1) clear it by using the up arrow to highlight "L1". Then press "CLEAR" and "ENTER". Enter your data into List 1 (L1) by entering each value and pressing "ENTER" after each value. Press the buttons "2nd" and "MODE" (QUIT) to signify the end of the data. Press the "STAT" button. Use the right arrow to highlight the "CALC" selection. Choose "1:1-Var Stats" by pressing "ENTER". Press "ENTER" again. Use the down arrow to scroll the remaining statistics on the screen. The mean uses the symbol x. Press "CLEAR" to clear the screen.

Using Excel to find statistics on data: Step by step: Open an Excel worksheet and paste or import a data set into a column. Click on the menu “Data”, then the tab “Data Analysis”, and choose “Descriptive Statistics” In “Input Range” enter the range or highlight your data column. In “Output Range” click on a remote cell where you want the output to appear. Check “Summary Statistics” box Click “OK”. The output will consist in several statistics, including: Mean, Median Mode, Minimum, Maximum, and Count

Exercise: (continued) SUV price data: $14,655 $14,799 $15,605 $16,395 $16,798 $17,990 $19,300 $20,000 $21,995 $22,195 $22,708 $23,240 $23,405 $23,920 $25,176 $25,999 $26,185 $26,268 $27,815 $27,910 $28,680 $28,950 $29,099 $29,249 $30,585 $30,645 $31,985 $32,250 $32,950 $33,595 $33,790 $34,590 $35,550 $36,300 $38,175 $41,188 $42,660 $54,950 $56,000 $63,500 Find the Mean and Median of these prices. Answers: Using the calculator: Mean = $29, Median: Find the position (n+1)/2 = (40+1)/2 = 20.5 The median is the value halfway between the 20th and 21st values. The 20th value is $27,910 and the 21st value is $28,680. Median = ($27,910 + $28,680)/2 = $28,295