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Describing distributions with numbers William P. Wattles Psychology 302 1 Measuring the Center of a distribution Mean – – Median – The arithmetic average Requires measurement data The middle value Mode – The most common value 2 Measuring the center with the Mean x X n 3 Our first formula X the mean X the individual score n the number of individuals sum of 4 The Mean One number that tells us about the middle using all the data. The group not the individual has a mean. 5 Population Sample Sample mean X 6 Mu, the population mean 7 Population Sample X Calculate the mean with Excel Save the file psy302 to your hard drive – right click on the file – save to desktop or temp Open file psy302 Move flower trivia score to new sheet 8 Calculate the mean with Excel Rename Sheet – Calculate the sum – type label: total Calculate the mean – double click sheet tab, type flower type label: mean Check with average function 9 Measuring the center with the Median Rank order the values If the number of observations is odd the median is the center observation If the number of observations is even the median is the mean of the middle two observations. (half way between them) 10 Measuring the center with the Median n 1 Median 2 11 The mean versus the median The Mean – – uses all the data has arithmetic properties The Median – less influenced by Outliers and extreme values 12 Mean vs. Median Betty Mike Tom Miriam Stacy David Mary Lou John Gail Arthur $ 28,514.00 $ 22,316.00 $ 30,112.00 $ 29,521.00 $ 21,555.00 $ 125,366.00 $ 22,132.00 $ 27,561.00 $ 24,635.00 $ 30,125.00 mean median $ 36,183.70 $ 28,037.50 The Mean The mean uses all the data. The group not the individual has a mean. We calculate the mean on Quantitative Data Three things to remember 5 The mean tells us where the middle of the data lies. We also need to know how spread out the data are. Measuring Spread Knowing about the middle only tells us part of the story of the data. We need to know how spread out the data are. Variability Variety is the spice of life Without variability things are just boring exam3 Psy314 Health Psychology 69% 61% 79% 100% 54% 60% 85% 83% 58% 75% 85% 73% 87% 57% 80% 83% 65% 68% 58% 50% 83% 55% 59% 79% 89% 74% 85% 63% Why is the mean alone not enough to describe a distribution? Outliers is NOT the answer!!!! The mean tells us the middle but not how spread out the scores are. 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Example of Spread New York mean annual high temperature 62 14 Example of Spread San Francisco mean annual high temperature 65 14 Example of Spread New York mean max min range 62 84 39 45 San Francisco 65 73 55 18 sd 17.1 6.4 16 Example of Variability Psy 302 Spring 2003 120% 100% Grade 80% Final 60% Quiz 7 40% 20% 0% 1 2 3 4 5 6 7 8 9 10 11 12 13 Student 14 15 16 17 18 19 20 21 22 23 Measuring Spread Range Quartiles Five-number summary – Minimum – first quartile – median – third quartile – Maximum Standard Deviation 17 Mean 50.63% Std Dev 21.4% Mean 33.19% Std Dev 13.2% Deviation score Each individual has a deviation score. It measures how far that individual deviates from the mean. Deviation scores always sum to zero. Deviation scores contain information. – How far and in which direction the individual lies from the mean 19 Measuring spread with the standard deviation Measures spread by looking at how far the observations are from their mean. The standard deviation is the square root of the variance. The variance is also a measure of spread 18 Mean X $28,756 $ 32,092 The average teacher $28,756, John $32.092 64.5 68 The average woman is 5 4 1/2, mary is 5'8" 110 90 The average IQ is 110 and Bubba has a 90 deviation score Individual deviation scores deviation score The average teacher $28,756, John $32.092 $3,336 dollars The average woman is 5 4 1/2, mary is 5'8" 3.5 inches The average IQ is 110 and Bubba has a 90 -20 points Standard deviation One number that tells us about the spread using all the data. The group not the individual has a standard deviation. Note!! Standard Deviation s ( x x ) 2 n 1 23 Variance s 2 ( x x ) 2 n 1 22 Properties of the standard deviation s measures the spread about the mean s=0 only when there is no spread. This happens when all the observations have the same value. s is strongly influenced by extreme values 24 New Column headed deviation Deviation score = X – the mean Calculate Standard Deviation with Excel In new column type heading: dev2 Enter formula to square deviation Total squared deviations – type label: sum of squares Divide sum of squares by n-1 – type label: variance 25 Moore page 50 Example 2.7page 50 subject subject1 subject2 subject3 subject4 subject5 subject6 subject7 MetabolicRate 1792 1666 1362 1614 1460 1867 1439 Example 2.6page 42 subject subject1 subject2 subject3 subject4 subject5 subject6 subject7 total mean MetabolicRate dev 1792 1666 1362 1614 1460 1867 1439 11200 1600 dev2 192 66 -238 14 -140 267 -161 36864 4356 56644 196 19600 71289 25921 0 214870 35811.6667 189.239707 189.239707 ss var stdev stdev check To Calculate Standard Deviation: Total raw scores divide by n to get mean calculate deviation score for each subject (X minus the mean) Square each deviation score Sum the deviation scores to obtain sum of squares Divide by n-1 to obtain variance Take square root of variance to get standard deviation. Population Sample Sample variance s 2 26 Population variance 2 27 Population Variance 2 Sample Variance s 2 Little sigma, the Population standard deviation 28 Sample standard deviation s 29 Population Standard Deviation Sample Standard Deviation s To analyze data 1. Make a frequency distribution and plot the data Look for overall pattern and outliers or skewness Create a numerical summary: mean and standard deviation. Start with a list of scores Cathy Paula Sandy Lois Anne Miriam June David 400 300 500 400 500 600 400 500 Alice Mitzi Jack Mike Dawn Vicki George Ashley 300 200 700 500 600 400 500 800 41 Make a frequency distribution 200 xxxxxxxxxxx 300 xxxxxxxxxxxxxxxxxxxxxxxx 400 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 500 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 600 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 700 xxxxxxxxxxxxxxxxxxxxxxxxx 800 xxxxxxxxxxxxx 42 Frequency distribution score 200 300 400 500 600 700 800 frequency 50 100 150 250 150 100 50 43 Represent with a chart (histogram) SAT Scores 250 150 100 50 800 700 600 500 400 300 0 200 Frequency 200 Score 44 Represent with line chart SAT Scores 250 150 100 50 800 700 600 500 400 300 0 200 Frequency 200 Score 45 Density Curve Replaces the histogram when we have many observations. Transform a score Hotel Atlantico 200 pesos Peso a unit of measure Transform a score 1 dollar = 28.38 pesos 200/28.38=$7.05 Dollar a unit of measure standardized observations or values. To standardize is to transform a score into standard deviation units. Frequently referred to as z-scores A z-score tells how many standard deviations the score or observation falls from the mean and in which direction 31 Standard Scores (Z-scores) individual scores expressed in terms of the mean and standard deviation of the sample or population. Z = X minus the mean/standard deviation 32 Z-score z x 33 new symbols the population mean the population s tan dard deviation z the s tan dardized value 34 Calculate Z-scores for trivia data Label column E as Z-score Type formula deviation score/std dev Make std dev reference absolute (use F4 to insert dollar signs) Copy formula down. Check: should sum to zero 35 File extensions Word .doc Excel .xls Text files .txt To view File extensions Open Windows Explorer Choose Tools/Folder Options/View uncheck “hide extensions for known file types. Z Scores Height of young women – – Mean = 64 Standard deviation = 2.7 How tall in deviations is a woman 70 inches? A woman 5 feet tall (60 inches) is how tall in standard deviations? 37 Z scores Height of young women – – Mean = 64 Standard deviation = 2.7 How tall in deviations is a woman 70 inches? z = 2.22 A woman 5 feet tall (60 inches) is how tall in standard deviations? z = -1.48 38 Calculating Z scores height 70 64 z 2.22 2.7 z height 60 64 1.48 2.7 39 Calculating X from Z scores X z * Types of data Categorical or Qualitative data – Nominal: Assign individuals to mutually exclusive categories. exhaustive: everyone is in one category – Ordinal: Involves putting individuals in rank order. Categories are still mutually exclusive and exhaustive, but the order cannot be changed. 72 Types of data Measurement or Quantitative Data – Interval data: There is a consistent interval or difference between the numbers. Zero point is arbitrary – Ratio data: Interval scale plus a meaningful zero. Zero means none. Weight, money and Celsius scales exemplify ratio data – Measurement data allows for arithmetic operations. 73 Review Video2 The End 60 Mean vs. Median