The variable: scores on a 60 question exam for 20 students 50, 46, 58, 49, 50, 57, 49, 48, 53, 45, 50, 55, 43, 49, 46, 48, 44, 56, 57, 44

Transcription

1 Cal State Northridge Ψ Andrew Ainsworth PhD The variable: scores on a question exam for students,, 8, 9,, 7, 9, 8,,,,,, 9,, 8,,, 7, First Step Order the Data,,,,,, 8, 8, 9, 9, 9,,,,,,, 7, 7, 8 1

2 Valid Total Cumulative Frequency Percent Valid Percent Percent When sets of data become very large with a large number of response categories (e.g. continuous data) it is sometimes easier to see a clear pattern in the data by grouping them into class intervals. One can then form a Grouped Frequency Distribution, especially if the data are assumed to be continuous.

3 Construct classes of data, where number of classes varies between 1 (depending upon the range of scores). Size of the class interval is: X highest X lowest i = chosen number of intervals For our example: 8 i = = 1. - We'll round off to 1 7 Apparent Real Relative Cumulative Cumulative Relative Limits Limits Midpoint Frequency Frequency Frequency Frequency rf =f/n, e.g.,1/ =. crf =cf/n 8 Class interval: - Class interval: -9 Class interval: - Upper Stated Limit Lower Stated Limit Upper Stated Limit Lower Stated Limit Lower real limit (-9 interval) Midpoint Lower real limit (- interval) Upper real limit (- interval) Upper real limit (-9 interval) 9

4 Apparent Class Limits Real Class Limits 1,-,,-, 1,-, 1,-, 11, -1, 1,-1, Unit of difference = Level of Accuracy If the smallest unit of measurement is $1, this is the level of accuracy/unit of difference,-1,,-1, 1,-, -, Real lower limit = apparent lower limit -.(unit of difference) Real upper limit = apparent upper limit +.(unit of difference) Class interval = i = Real upper limit real lower limit (,,=,) 1 Income rounded to $1, Income rounded to $1 Apparent Class Limits Real Class Limits 1,-,,-, 1,-, 1,-, 11, -1, 1,-1,,-1,,-1, 1,-, -, Apparent Class Limits Real Class Limits,1-,,-, 1,1-, 1,-, 1,1-1, 1,-1,,1-1,,-1, 1-, -, Person earning $,1 Nature of distribution will also depend upon number of classes used 11 Histograms Frequency Polygons Bar Graphs Pie-charts Stem & Leaf plots 1

5 classes Classes classes Height of bar = # of responses in the interval Width of bar = size of the interval Bars touch representing grouped continuous data 1

6 Frequency Polygon Frequency Test Score Interval 1 MARITAL STATUS Percent 1 MARRIED DIVORCED NEVER MARRIED WIDOWED SEPARATED MARITAL STATUS 17 Like Histograms The height indicates the frequency Unlike Histograms Bars represent categories Width is Meaningless Bars DO NOT touch Discrete Data 18

7 MARITAL STATUS Missing.1% NEVER MARRIED 19.1% SEPARATED.7% MARRIED DIVORCED.% 1.% WIDOWED 11.% 1. Pie-charts are especially good when showing distributions of a few qualitative classes and one wishes to emphasize the relative frequencies that fall into each class.. However, not as effective with a) large number of classes. b) with numerical data because the circle is confusing when ordered classes are represented. 19 A stem and leaf diagram provides a visual summaryof your data. This diagram provides a partial sortingof the data and allows you to detect the distributional pattern of the data. There are three steps for drawing a tem and leaf diagram. 1. Split the data into two pieces, the stem(left 1,, digits, etc.) and the leaf(the right most digit).. Arrange the stems from low to high.. Attach each leaf to the appropriate stem. Ordered Testscore Data,,,,,, 8, 8, 9, 9, 9,,,,,,, 7, 7, 8 What are the stems? What are the leaves? 1 7

8 Stem-and-Leaf Plot Frequency Stem & Leaf Stem width: 1 Each leaf: 1 case(s) Here the stem width is ten because the stems represent numbers in the 1s place numerically Advantage: Combines frequency distribution with histogram, thereby giving a pictorial description of data. Disadvantages: Only works with numerical data. Works best with small and compact data sets (e.g., will not work well with 1, cases & data in the range of -). Modality ( How many peaks are there?) Unimodal, bi-modal, multimodal Symmetric vs. Skewed Skewed positive (floor effect) Skewed negative (ceiling effect) Kurtosis ( How peaked is your data? ) Leptokurtic, Mesokurticand Platykurtic 8

9 T erm s tha t D escrib e D istribu tio ns T e rm F eatu res E xa m p le left side is m irror "S ym m etric" im age of right side "P ositively skew ed " right tail is longer then the left "N egatively skew ed " left tail is longer than the right "U nim odal" one highest po int "B im odal" tw o high points "N orm al" unim odal, sym m etric, asym p totic 9