PSY 307 – Statistics for the Behavioral Sciences Chapter 2 – Describing Data with Tables and GraphsClass Progress To-Date Math Readiness Descriptives Midterm next MondayFrequency Distributions  One of the simplest forms of measurement is counting  How many people show a characteristic, have a given value or are members of a category.  Frequency distributions count how many observations exist for each value for a particular variable.Frequency Table  A frequency table is a collection of observations:  Sorted into classes  Showing the frequency for each class.  A “class” is a group of observations.  When each class consists of a single observation, the data is considered to be ungrouped.Creating a Table  List the possible values.  Count how many observations exist for each possible value.  One way to do this is using hash-marks and crossing off each value.  Figure out the corresponding percent for each class by dividing each frequency by the total scores.Unorganized Data 1, 5, 3, 3, 6, 2, 1, 5, 2, 1, 2, 6, 3, 4, 1, 6, 2, 4, 4, 2  A set of observations like this is difficult to find patterns in or interpret.ExampleWhen to Create Groups  Grouping is a convenience that makes it easier for people to understand the data.  Ungrouped data should have <20 possible values or classes (not <20 scores, cases or observations).  Identities of individual observations are lost when groups are created.Guidelines for Grouping  See pgs 29-30 in text.  Each observation should be included in one and only one class.  List all classes, even those with 0 frequency (no observations).  All classes with upper & lower boundaries should be equal in width.Optional Guidelines  All classes should have an upper and lower boundary.  Open-ended classes do occur.  Select an interval (width) that is natural to think about:  5 or 10 are convenient, 13 is not  The lower boundary should be a multiple of class width (245-249).  Aim for a total of about 10 classes.Gaps Between Classes  With continuous data, there is an implied gap between where one boundary ends and the other starts.  The size of the gap equals one unit of measurement – the smallest possible difference between scores.  That way no observations can ever fall within that gap.  Class sizes account for this.Relative Frequency  Relative frequency – frequency of each class as a fraction (%) of the total frequency for the distribution.  Relative frequency lets you compare two distributions of different sizes.  Obtain the fraction by dividing the frequency for each group by the total frequency  Total = 1.00 (100%)Example Total = 20 4/20 = .20 or 20% 5/20 = .25 or 25% 3/20 = .15 or 15% 3/20 = .15 or 15% 2/20 = .10 or 10% 3/20 = .15 or 15% Total = 1.0 or 100%Cumulative Frequency  Cumulative frequency – the total number of observations in a class plus all lower-ranked classes.  Used to compare relative standing of individual scores within two distributions.  Add the frequency of each class to the frequencies of those below it.Relative Frequency (Percent) and Cumulative FrequencyCumulative Proportion (Percent)  The cumulative proportion or percent is the relative cumulative frequency.  Percent = proportion x 100  It allows comparison of cumulative frequencies across two distributions.  To obtain cumulative proportions divide the cumulative frequency by the total frequency for each class.  Highest class = 1.00 (100%)Percentile Ranks  Percentile rank – percent of observations with the same or lower values than a given observation.  Find the score, then use the cumulative percent as the percentile rank:  Exact ranks can be found from ungrouped data.  Only approximate ranks can be found from grouped data.Qualitative Data  Some categories are ordered (can be placed in a meaningful order):  Military ranks, levels of schooling (elementary, high school, college)  Frequencies can be converted to relative frequencies.  Cumulative frequencies only make sense for ordered categories.Interpreting Tables  First read the title, column headings and any footnotes.  Where do the data come from, source?  Next, consider whether the table is well-constructed – does it follow the grouping guidelines.  Finally, look at the data and think about whether it makes sense.  Focus on overall trends, not details.Parts of a GraphConstructing Graphs  Select the type of graph.  Place groups on the x-axis.  Place frequency on the y-axis.  Values for the groups and frequencies depend on the data.  Label the axes and give a title to the graph.Histograms  For quantitative data only.  Equal units across x axis represent groups.  Equal units across y axis represent frequency.  Use wiggly line to show breaks in the scale.  Bars are adjacent – no gaps.Histogram Applets  http://www.stat.sc.edu/~west/javahtml/Histogram.html  Uses Old Faithful geyser data  http://www.shodor.org/interactivate/activities/histogram/?version=1.6.0_11&browser=MSIE&vendor=Sun_Microsystems_Inc.  Uses math SAT data  Notice that “bin width” refers to class or interval size.  SPSS automatically creates classes or intervals.Frequency Polygons  Also called a line graph.  A histogram can be converted to a frequency polygon by connecting the midpoints of the bars.  Anchor the line to the x axis at beginning and end of distribution.  Two frequency polygons can be superimposed for comparison.Creating a Line Graph from a Histogram Frequency Polygon0123456780 2 4 6 8 10 12Years of ServiceNumber of EmployeesHistogram012345670 2 4 6 8 10 12Length of Service (years)Number of EmployeesHistogram012345670 2 4 6 8 10 12Length of Service (years)Number of EmployeesStem-and-Leaf Displays  Constructing a display:  Notice the highest and lowest 10s  Arrange 10s in ascending order.  Copy right-hand digits as leaves.  The resulting display resembles a frequency histogram.  Stems are whatever digits make sense to use.Sample Stem and leaf display showing the number of passing touchdowns. 3|2337 2|001112223889 1|2244456888899Purpose of Frequency Graphs  In statistics, we are interested in the shapes of distributions because they tell us what statistics to use.  They let us identify outliers that might distort the statistics we will be using.  They present data so that readers can quickly