Frequency Distributions Chapter 3 of Understanding Statistics in the Behavioral Sciences by Pagano

Review Quiz (not on videos) 30 Honda USA employees are randomly chosen in a study of the stress experienced by Honda USA employees. The average stress level of the 30 employees is 7.8. The average age of the Honda USA employees is 36.o. The average age of Ford employees is 38.2. A. What is the sample? B. What is the population being studied? C. What is the relevant statistic?

Chapter 3: Frequency Distributions Example: Results from a pre-employment test It’s really hard to tell what’s going on.

Frequency Distributions A simple frequency distribution: Class interval (bin width) = 1 We can find the range (high and low scores), etc. The distribution still looks kind of choppy (too many empty bins)

Grouping the scores in a frequency distribution We can use a bin (a class interval) with a width of 2 (or more)

Making a frequency distribution

Shoe Size Exercise Make a Frequency Distribution for Shoe Sizes of a sample of APU Students. What size Bin Widths should we use? Use round numbers or logical divisions to define the bins Finger rule: 5-10 bins is good. Avoid empty bins Process Define the bins Tally Count Data: 8 8.5 9.5 8.5 6.5 11 7 6 6.5 9 9 9.5 8.5 10.5 8 8 7

Shoe Size Exercise One possible frequency distribution:

Relative and Cumulative Frequency

Frequency Tables in Excel Open Chapter 3 Data on 100 employees.xlsx Let’s add a column and employee ID Select Column A Right click, Insert Add column header “ID” Enter 1 and 2 in the first two rows. Select the cells with 1 and 2 in them. Drag to continue count.

Frequency Tables in Excel Create table from lumber company data Click in any cell in the table Insert => Table Note: Any new data you add to a table will be included in the table. Create a Pivot Table A pivot table counts, sorts, totals, or averages data from one table and puts in another table. Put curser in table Table Tools/Design => Summarize with PivotTable Choose new worksheet (tab) or existing worksheet

Frequency Tables in Excel Create a Frequency Table for employee age. Frequency tables: Bins go in rows: Drag Age into Row Labels Columns indicate the variable and the count Select Age Drag it to Row Labels Drag it to Summation Values Change to Count (of Age) Value Field Settings => Count

Frequency Tables in Excel Adjust column headings Age, Frequency (f) Put data into useful bins Select any age. Right click => group Adjust groups Set bin width to 10 Start at 10 Readjust first group

Frequency Tables in Excel Example 2: Frequency Table of Education Table Tools=> Summarize with PivotTable Rows: Education, Σ Values = Count of Educ. Note that we can’t tell where people with a high school education go (12 years of education). The real limits of the bin are 10.00 – 11.99 We need to change the bin labels to something meaningful Education is measured in half year steps. Years of Education Frequency 8-10 1 10-12 11 12-14 62 14-16 4 16-18 14 18-20 8 Grand Total 100 Years of Education Frequency 8-9.5 1 10-11.5 11 12-13.5 62 14-15.5 4 16-17.5 14 18 8 Grand Total 100

Frequency Table Exercise Exercise: Make a frequency table of something that interests you from the class data set. Click in table, Table Tools => Summ with PivotTable Put in Existing Worksheet. Choose variable for rows, set Value Field to Count Group so that all bins are included, even when f = 0. If there are empty bins, make them show: Pivot Table Field List => Row Labels => Field Settings =>Layout & Print, select “Show items with no data” Relabel your bins, if necessary, so that they are clear.

Percentiles The 75th percentile (P75)is the point on a scale that 75% of values fall below. Example: Height (N = 4) X1 =64 inches X2 =62 X3 =61 X4 =60 75% of the values would be below 63. Pages 47-49 (8e)/51-54 (9e)/56-58 (10e) give one of several complicated way of computing percentile points. Manually, it’s easier and more intuitive to put the scores in order and just count, getting an approximate value.

Calculating Percentiles on Excel You want to increase the work hours of 5% of your employees. You want to help out those who work the least. What is the 5th percentile of the number of hours worked? (use Chapter 3 Data on 100 Employees.xls) Order the data (Usual Hours Worked) from high to low. Find the value below which are 5% of the employees. Select appropriate column. Home => Conditional Formatting => Top/Bottom Rules => Bottom 10 % => 5% => OK: 35 would be the 5th percentile.

Calculating Percentiles on Excel What is the percentile rank of someone who works 52 hours per week? Start a new column “Percentile Rank of Hours”. In the first cell, use fx to find the percentrank.exc function Percentrank.exc(array, x, signficance) Array: the list of numbers we’re considering ($B$4..$B$103) x = the location of the first score (B4) Significance = number of significant figures (not impt; we’ll change it) Change entire column to percentage, adjust number of decimals

Making Graphs of Frequency Distributions Every year a school has about 19,900 students. Proper placement of the origin on the y (vertical) axis Data 1993: 19,800 1994: 19,980 1995: 20,000 1996: 19,960 etc.

Making Graphs of Frequency Distributions Improper placement of the origin on the y (vertical) axis

The Bar Graph (Bar Chart) A plot of the frequency distribution for nominal (or possibly ordinal) data. => Clearly label the x-axis! The height of the bar represents the frequency. Space is shown between data.

Bar Chart/Bar Graph Example: Who Uses a Building? Daily users of a building? 14 faculty 21 administrative 9 support Draw a bar chart: Create a two column frequency table: 1) Users, 2) Count Place curser in single cell in table. Insert => Chart/Column => 2-D/Clustered Column Adjust title: Daily Users of Building Click on legend to remove it.

Bar Graph/Bar Chart Exercise Create a bar chart of the 100 Employees’ sex Work as a class before working in groups with another variable. Remember to start by creating a table, and then a pivot table Mac Users: If you're having trouble making a bar chart or histogram from a pivot table, select the whole pivot table except the last line labeled Grand Total. Then go to Insert and choose a chart (2-D Bar Column).

The Histogram A plot of the frequency distribution for ordinal, interval, or ratio data The height of the bar represents the frequency (be sure to label both axes). The bins (class intervals) are on the x-axis with no space in between. Tic marks on x-axis with one of the following: -Bin ranges -Bin mid-points -End-points of real limits

Histograms in Excel Make a frequency distribution pivot table with the bins labeled usefully (e.g., frequency table for age) Click in pivot table for age Insert => Chart/Column => 2-D Column/Clustered Column) Adjust to make histogram Double click a bar => Change gap width to 0% (or Layout 8 under Design) Change boarder color to “solid line” Double click on title and change to “Histogram of Age” Click on legend and delete it.

Frequency Polygons and Curves A frequency polygon is like a histogram, but the bars are replaced with dots at mid-point of the class interval. Smoothed out, we call it a frequency curve Frequency Polygon

Shapes of Frequency Curves

Why a Bell Curve? 1. Human behavior, thought, etc. is varied. => Scores spread out 2. There tends to be a normal or most common value for whatever we measure in humans. = > The high point 3. Some values are above average and some values are below average, but most are values are close to average => Frequency distributions tail off gradually in both directions. Note: Skew often comes from “ceiling” and “floor” effects.