FREQUENCY DISTRIBUTIONS Descriptive Statistics Overview Frequency distributions tables

FREQUENCY DISTRIBUTIONS Descriptive Statistics

Overview Frequency distributions & tables Relative & cumulative frequencies/percentages Graphs

Describing data On a 7 pt scale with anchors of 1 (very easy) and 7 (very difficult), how difficult do you think this class is? 7 4 6 2 5 3 5 3 6 5 6 6 1 6 5 3

Frequency Distribution Table Scores 7 6 5 4 3 2 1 f 1 5 5 2 3 2 1 Sf = N N? Maximum score? Minimum score? Range? Scores cluster? Spread of scores?

Constructing a Frequency Table Scores 7 6 5 4 3 2 f 1 0 3 1 3 2 Scores listed from high to low � Note: can do it the other way, but it will be easier to check answers if everyone does it the same way List all possible scores between highest & lowest, even if nobody obtained such score Notice f is in italics

Example Gender Men Women f 24 7 N = 31 Notice: This is on a nominal scale.

Simple Frequency Distribution 7 Can find ΣX using frequency distribution: � Multiply each X by f, then sum Scores f 10 2 9 5 8 7 7 3 6 2 5 0 4 1 ΣX = f*Scores

Simple Frequency Distribution 8 Finding ΣX Scores f 10 2 20 9 5 45 8 7 56 7 3 21 6 2 12 5 0 0 4 1 4 ΣX = 158 f*Scores

Grouped frequency distribution Scores grouped into intervals & listed along with the frequency of scores in each interval � Guidelines: Non-overlapping intervals, 10 -20 intervals, widths of intervals should be simple (e. g. , f rel. f cf percenti 5, Score 10) le 40 -44 2 . 08 25 100 35 -39 2 . 08 23 92 30 -34 0 . 00 21 84 25 -29 3 . 12 21 84 20 -24 2 . 09 18 72 15 -19 4 . 16 16 64 10 -14 1 . 04 12 48 5 -9 4 . 16 11 44 0 -4 7 . 28 7 28 Note: We lose info about specific values.

Relative & Cumulative Frequencies

Relative Frequency Relative frequency (rel. f or rf) rel. f = Score f rel. f 6 1 . 05 5 0 . 00 4 2 . 10 3 3 . 15 2 10 . 50 1 4 . 20

Relative Frequency Score 12 11 10 9 8 7 N= 3 2 5 20 f rel. f. 15 (15% of the class received a score of 12). 10. 25 (25% of the class received a score of 10). 15. 10. 25

Cumulative Frequency of all scores at or below a particular score Score f cf 17 1 20 16 2 19 15 4 17 14 6 13 13 4 7 12 0 3 11 2 3 10 1 1

Cumulative Frequency Distribution Score 12 11 10 9 8 7 N= 3 2 5 20 f 20 17 15 10 7 5 cf (17 people scored at or below 11) (10 people scored at or below 9) (5 people scored at or below 7)

Cumulative % Percent of all scores in the data that are at or below the score Cumulative % =

Practice 1 Using the following data set: � � Create a simple distribution table - find the relative frequency, find the cumulative frequency, and find the cumulative percent for each remaining data points Note: Round to the 2 nd decimal place 5 4 3 5 1 1 4 3 3 1 4 4

Practice 1: Answers Scores f rf cf c% 5 2 . 17 12 100% 4 4 . 33 10 83% 3 3 . 25 6 50% 2 0 . 00 3 25% 1 3 . 25 3 25%

Practice 2 Using the following data set: � � Create a simple distribution table - find the relative frequency, find the cumulative frequency, and find the cumulative percent for each remaining data points Note: Round to the 2 nd decimal place 2 5 5 2 8 8 8 6 6 4 7 8 6

Practice 2: Answers scores f rf cf c% 8 4 . 31 13 100% 7 1 . 08 9 69% 6 3 . 23 8 62% 5 2 . 15 5 38% 4 1 . 08 3 23% 3 0 . 00 2 15% 2 2 . 15 2 15%

Graphs

Graphs X axis – horizontal (scores increase from left) Y axis – vertical (scores increase from bottom) Scale of measurement determines type of graph � Bar graph � Histogram � Polygon

Bar Graphs Spaces between bars � Distinct categories Used with nominal scales or qualitative data � Sometimes also used with ordinal scales

Histograms No spaces between bars Labels directly under each box Used with ordinal, interval, or ratio scales Usually used with discrete data

Polygons Used when � larger range of scores � Interval or ratio scales � Continuous data Dot centered above each score if it is discrete data

“Most misleading graph ever published”

Distributions Normal curve

Variations in Distributions Kurtosis = how peaked or flat distribution Mesokurtic = normal Leptokurtic = thin Platykurtic = broad/ fat

Variations in Distributions Negatively skewed (left skew)

Variations in Distributions Positively skewed (right skew)

Variations in Distributions Bimodal

Practice 3 Using the following data set: � � Create a simple distribution table, find the relative frequency, find the cumulative frequency, and find the cumulative percent for each remaining data point --- also, create a graph. Note: Round to the 2 nd decimal place (assume interval scale) 14 14 13 15 11 15 13 10 12 13 14 15 17 14 14 15

Practice 3: Histogram 7 6 Frequency 5 4 3 2 1 0 10 11 12 13 14 Scores 15 16 17

Practice 3: Answers X f rf cf c% 17 1 . 06 18 100% 16 0 . 00 17 94% 15 4 . 22 17 94% 14 6 . 33 13 72% 13 4 . 22 7 39% 12 1 . 06 3 17% 11 1 . 06 2 11% 10 1 . 06 1 06%