Introduction to Statistics in Psychology DISTRIBUTION USES LIMITATIONS summarize data Last time we discussed percentiles and percentile ranks PSY 201 Professor Greg Francis Lecture 05 central tendency Wanna bet indicate most frequent data values indicate amount of variation across data values allows us to interpret a single score in the context of other scores we are exploring quantitative methods to describe distributions very useful for comparing a score to a distribution of scores not so good for talking about a distribution overall want to quantify ideas of central tendency most of scores average score mode median mean and variation how variable scores are in a distribution 2 3 MODE MODE MODE the most frequent data value score top of a hill on a frequency distribution graph we actually look for a modal interval and consider the midpoint of the interval to be the mode easy to find from a table of frequency scores 4 easy to find unimodal distribution when there is a single mode single hill multimodal distribution when there are several modes many hills 40 30 Frequency Exact Limits Midpoint f cf c 64 5 69 5 67 6 180 3 33 100 59 5 64 5 62 15 174 8 33 96 67 54 5 59 5 57 37 159 20 56 88 34 49 5 54 5 52 30 122 16 67 67 78 44 5 49 5 47 42 92 23 33 51 11 39 5 44 5 42 22 50 12 22 27 78 34 5 39 5 37 18 28 10 00 15 56 29 5 34 5 32 7 10 3 89 5 56 24 5 29 5 27 2 3 1 11 1 67 19 5 24 5 22 1 1 0 56 0 56 bimodal distribution when there are two modes two hills 20 10 0 25 30 35 40 45 50 55 60 65 70 NOTE the use of the terms are not quite consistent Score 5 6 BIMODAL MEDIAN CALCULATIONS this distribution might be called bimodal even though there is really only one mode the point below which 50 of scores fall for our data set the 50th percentile Mdn P50 ll 40 n 0 5 cf w fi ll 44 5 exact lower limit of the interval containing the n 0 5 score n 180 total number of scores 0 5 50 100 proportion corresponding to 50th percentile decimal form cf 50 cumulative frequency of scores below the interval containing the n 0 5 score 100 20 80 10 0 25 30 35 40 45 50 55 60 65 70 PERCENTAGE Frequency 30 60 fi 42 frequency of scores in the interval containing the percentile point 40 Score 20 not very useful for mathematics w 5 width of class interval 0 25 30 35 40 45 50 55 60 65 70 Score Mdn P50 44 5 7 8 9 CALCULATIONS CALCULATIONS MEAN when the raw scores are used instead of class intervals 1 Arrange the scores in ascending order from lowest to highest 2 If there is an odd number of scores the median is the middle score 3 If there is an even number of scores the median is halfway between the two middle scores Will this always give the same value as for the frequency distribution approach Name Aimee Greg Ian Jim Sex Female Male Male Male Score 94 95 89 92 scores 89 92 94 95 even number of scores the median is halfway between 92 and 94 93 Name Aimee Greg Ian Jim Bob Sex Female Male Male Male Male Score 94 95 89 92 83 scores 83 89 92 94 95 odd number of scores arithmetic average of scores in a distribution mean of a population is designated as mean of a sample is designated as X Calculated as 1 n X X n i 1 i Xi the ith score n total number of scores sometimes just written as 1 X Xi n the median is the middle score 92 10 11 180 0 5 50 5 49 26 42 12 CALCULATIONS Name Aimee Greg Ian Jim Sex Female Male Male Male X Score 94 95 89 92 1 Xi n 1 X X1 X2 X3 X4 4 1 370 X 95 89 94 92 92 5 4 4 COMPARISON mean can only be used on interval or ratio data mode and median can be used on any data mean can be manipulated mathematically mean can be sensitive to extreme scores suppose a manufacturing company is getting a tax break from the local government the tax break is up for renewal and the company argues it should continue because it provides an average salary of 47 684 for 38 people a local group opposed to the renewal points out that the modal salary is 22 000 COMPARISON Here s the data for the company Position Number Salary President 1 500 000 Ex vice pres 1 150 000 Vice pres 2 120 000 Controller 1 60 000 Senior sales 3 45 000 Junior sales 4 36 000 Foreman 1 33 000 Machinists 12 22 000 If you just averaged the salaries you would get 120 750 but this is not correct a councilman notes that the median salary is also 22 000 they have all done the correct calculations 13 14 15 MEAN OF MEANS COMBINED GROUPS COMBINED GROUPS the mean of means is not the same thing as the mean of all the scores in all the groups correct calculation goes like n X nM X M X F F nF nM where X F is the mean for the females X M is the mean for the males nF is the number of females nM is the number of males Name Greg Ian Aimee Jim Sex Male Male Female Male X X X Score 95 89 94 92 Xi n X1 X2 X3 X4 4 95 89 94 92 370 92 5 4 4 Consider our small data set 94 94 0 1 95 89 92 XM 92 0 3 XF the mean of the means is XF XM 93 0 2 but we already found that the mean of all the scores was X 92 5 1 94 0 3 92 0 1 3 94 276 92 5 4 X same as direct calculation of X too much weight on the female group 16 17 18 COMBINED GROUPS WANNA BET WANNA BET in general given the properties of averages make for some odd conclusions that you can take advantage of over the second half of the season they get X i individual group means ni number of observations in individual groups N ni total number of observations in all groups ni X i X N batting averages number of hits X number of at bats consider two players who are competing for the batting title highest average over the first half of the season they get Jim At bats 45 Hits 12 Average 0 267 John 12 3 0 250 Jim At bats 15 Hits 5 Average 0 333 John 60 19 0 317 but over the entire season they get Jim At bats 60 Hits 17 Average 0 …