PSY 307 Statistics for the Behavioral Sciences Chapter 2 Describing Data with Tables and Graphs
Class Progress To-Date Math Readiness Descriptives Midterm next Monday
Frequency 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.
Example
When 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 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 = 20 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 Frequency
Cumulative 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 Graph
Constructing 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_microsys tems_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.
Number of Employees Number of Employees Number of Employees Creating a Line Graph from a Histogram 7 6 5 4 3 2 1 0 Histogram 0 2 4 6 8 10 12 Length of Service (years) 8 7 6 5 4 3 2 1 0 Frequency Polygon 0 2 4 6 8 10 12 Years of Service 7 6 5 4 3 2 1 0 Histogram 0 2 4 6 8 10 12 Length of S ervice (years)
Stem-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 2244456888899
Purpose 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 and easily grasp its meaning.
Shapes of Distributions Normal bell-shaped and symmetrical. Bimodal two peaks. Suggests presence of two different types of observations in the same data. Positively skewed lopsided due to extreme observations in right tail. Negatively skewed extreme observations in left tail.
Shapes of Graphs bimodal normal positive skew negative skew
Heavy vs Light-tailed Distributions Heavy-tailed a distribution with more observations in its tails. Light-tailed a distribution with fewer observations in its tails and more in the center. Kurtosis a statistic that measures the shape of the distribution and the size of the tails.
Other Kinds of Graphs Frequency is not the only measure that can be displayed on the y-axis. We are using a graph to explore the shape of a distribution in this chapter. Usually the y-axis shows the dependent variable while the x-axis shows groups (independent variable). Graphs can be visually interesting!
Graphs Allow Visual Comparisons
The Best Graph Ever Drawn Source: http://strangemaps.wordpress.com/
Details About the Graph The map was the work of Charles Joseph Minard (1781-1870), a French civil engineer who was an inspectorgeneral of bridges and roads, but whose most remembered legacy is in the field of statistical graphics The chart, or statistical graphic, is also a map. And a strange one at that. It depicts the advance into (1812) and retreat from (1813) Russia by Napoleon s Grande Armée, which was decimated by a combination of the Russian winter, the Russian army and its scorched-earth tactics. To my knowledge, this is the origin of the term scorched earth the retreating Russians burnt anything that might feed or shelter the French, thereby severely weakening Napoleon s army. It unites temperature, time, geography and number of soldiers, all in one picture.
A Modern Version
Qualitative Data Bar graphs similar to histograms. Bars do not touch. Categorical groups are on x-axis. Pie charts Where tax money goes.
Misleading Graphs Bars should be equal widths Bars should be two-dimensional, not three-dimensional When the lower bound of the y-axis (frequency) is cut-off (not 0), the differences are exaggerated. Height and width of the graph should be approximately equal.
Graphs are Used to Persuade Reagan Bush Clinton Bush
Gallup s Terry Schiavo Poll
Who Increased the Debt? This chart is misleading because it includes social security as debt. If expressed as a % of public debt, Bush & Obama would be tied around 60-70%. Obama would look 4 times worse than Bush and twice as bad as Reagan if this were expressed as a % of income (GDP).
Misleading Tables Average score, reading literacy, PISA, 2009: Korea 539 Finland 536 Canada 524 New Zealand 521 Japan 520 Australia 515 Netherlands 508 Belgium 506 Norway 503 Estonia 501 Switzerland 501 Poland 500 Iceland 500 United States 500 Sweden 497 Germany 497 Ireland 496 France 496 Denmark 495 United Kingdom 494 Hungary 494 OECD average 493 Portugal 489 Italy 486 Slovenia 483 Greece 483 Spain 481 Czech Republic 478 Slovak Republic 477 Israel 474 Luxembourg 472 Austria 470 Turkey 464 Chile 449 Mexico 425
How Big are Crime Rates? Source: http://www.npr.org/templates/story/story.php? storyid=5480227
How Many Groups (Categories)? This graph is misleading because income above 200k is broken into many sub-categories, making the 100-200k group look larger than higher income groups. How it would look if redrawn.
Comparing Scales (OK)
Misleading Scales The range of the scales for these two variables are too different to be compared visually without being misleading. The crossover point at 2004 disappears when the same range is used on both scales of the graph.
More Misleading Graphs http://www.coolschool.ca/lor/ama1 1/unit1/U01L02.htm