DISTRIBUTIONS Introduction to Statistics in Psychology As we saw last time a well drawn graph conveys a lot of useful information PSY 201 Professor Greg Francis What all that stuff on the SATs meant CUMULATIVE Frequency 150 100 50 0 35 40 45 50 55 60 65 70 Score 100 PERCENTAGE 80 60 We would like a quantitative method of describing distributions may not entirely avoid msinformation but at least the limitations will be identifiable 40 30 Frequency describing distributions 30 frequency versus score class interval from data set in book but a poorly drawn graph can mislead and confuse Lecture 04 25 FREQUENCY DISTRIBUTIONS 20 10 0 25 30 35 40 45 40 45 50 55 60 65 70 DISTRIBUTION USES 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 summarize data 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 will explore quantitative methods to describe distributions 70 Score 4 65 TABLE FORMAT 0 35 60 3 20 30 55 2 40 25 50 Score 5 6 PERCENTILE PERCENTILE CALCULATIONS point in a distribution at or below which a given percentage of scores is found what are the data values for the lowest 60 of the population find P60 using the above data set of scores several steps 1 number of scores making up 60 of student scores is written as Ppercentage 28th percentile is written as P28 99th percentile is written as P99 1 Find out how many data values make up 60 of the population 180 0 60 108 2 Find the lowest class interval in the cumulative frequency distribution that includes at least that many data values In general calculate n p where n is the size of the population number of scores and p is the percentage in decimal form 3 Estimate how far into the class interval you must go to reach exactly the percentile works for any percentage 7 8 9 CALCULATIONS CALCULATIONS CALCULATIONS 2 lowest class interval in the cf including 108 scores is with midpoint 52 so we know that the percentile is somewhere between 49 5 and 54 5 We want a more precise estimate estimate of percentile point 10 we need to know 100 width of class interval 5 frequency of scores in the class interval containing the percentile point 30 exact lower limit of class interval containing the percentile point 49 5 cf of scores below the class interval containing the percentile point 92 remaining number of scores in class interval containing the percentile point 108 92 16 11 80 PERCENTAGE 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 go into the interval the remaining unaccounted for percentage 60 40 20 0 25 30 35 40 45 Score 12 50 55 60 65 70 PX ll np cf w fi ll exact lower limit of the interval containing the percentile point n total number of scores p X 100 proportion corresponding to percentile decimal form cf cumulative frequency of scores below the interval containing the percentile point fi frequency of scores in the interval containing the percentile point w width of class interval PERCENTILE RANK OGIVE given a particular data value what percentage of data values are smaller plot cumulative frequency percentage against score class interval gives percentile rank e g given a score on a test what percentage of scores were lower sort of the reverse of percentile for a data value of 39 we write the percentile rank as P R39 Used on achievement tests CALCULATIONS X score for which percentile rank is to be determined cf cumulative frequency of scores below the interval containing the score X ll exact lower limit of the interval containing X w width of class interval containing X fi frequency of scores in the interval containing X n total number of scores 16 40 0 30 35 40 45 50 55 60 65 Score CALCULATIONS cf fi X ll w 100 n 60 25 14 80 20 13 P RX 100 15 LIMITATIONS cf fi X ll w 100 n 10 18 39 34 5 5 P R39 100 180 P R39 14 556 P RX 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 17 percentiles help describe a data value relative to its frequency distribution but they have some drawbacks percentiles use an ordinal scale equal differences in percentiles do not indicate equal differences in raw scores class intervals with higher frequency cover a broader range of percentiles steeper part of ogive 100 80 PERCENTAGE PERCENTAGE CALCULATIONS 60 40 20 0 25 30 35 40 45 Score 18 50 55 60 65 70 70 LIMITATIONS CONCLUSIONS NEXT TIME percentiles exaggerate differences in scores when lots of people have similar scores percentiles central tendancy mode median mean percentile ranks Wanna bet underestimate actual differences when lots of people have very different scores differences in percentiles should not be compared across different distributions only provide information on relative ranking of scores ordinal scale cannot be meaningfully averaged summed multiplied fixing these problems requires additional terms for describing distributions central tendency 19 20 21