Sample Distributions

Review Parameter vs. Statistic Parameter vs. Statistic Population and Sample Population and Sample Construct Construct Variables (IV and DV) Variables (IV and DV) Operationalization Operationalization Levels of measurement Levels of measurement Summation notation Summation notation What is the value of  (X+2) 2 where X{1, 3, 4, 5}What is the value of  (X+2) 2 where X{1, 3, 4, 5}

 By the end of today we will have three ways to describe a distribution Shape Shape SkewSkew KurtosisKurtosis Central Tendency Central Tendency MeanMean MedianMedian ModeMode Variability Variability Standard DeviationStandard Deviation VarianceVariance

What is the job of descriptive statistics?  To simplify and organize data

How Best to Accomplish This?  Among other techniques, a frequency distribution is helpful  A frequency distribution is an organized tabulation of the number of individuals located in each category (from a sample) on the scale of measurement.

Example of a Frequency Distribution Table XfXf

What other information can this type of table give us?  It can help us find  X  It gives an easy way to see proportions and percentages Definition: A proportion measures the fraction of the total group that is associated with each score. Definition: A proportion measures the fraction of the total group that is associated with each score. proportion = p = f/Nproportion = p = f/N A percentage gives essentially the same information just in a different form A percentage gives essentially the same information just in a different form Percentage = proportion(100) = p(100) = (f/N)(100) then just add the percent signPercentage = proportion(100) = p(100) = (f/N)(100) then just add the percent sign

Example X f  20 p = f/N 2/20 = /20 = /20 = /20 = /20 = /20 = /20 = 0.05 % = p(100) 10% 25% 35% 15% 10% 0% 5% %

What might be another way we would want to look at this information?  A frequency distribution graph is a picture of the information available in a frequency distribution table  What are the basic parts of any graph? X-axis X-axis Y-axis Y-axis Scale Scale Origin Origin

The Basic Types of Graphs (Data From a Sample)  Histograms A histogram lists the numerical scores (categories of measurement) along the x- axis and creates bars that extend up to the height on the y-axis which represents the frequency for that category A histogram lists the numerical scores (categories of measurement) along the x- axis and creates bars that extend up to the height on the y-axis which represents the frequency for that category

Example of a Histogram

Basic Graphs Continued  Bar Graphs A bar graph is essentially the same as a histogram, except that spaces are left between adjacent bars. This space emphasizes that the scale is either nominal, or ordinal not interval or ratio. A bar graph is essentially the same as a histogram, except that spaces are left between adjacent bars. This space emphasizes that the scale is either nominal, or ordinal not interval or ratio.

Example of a Bar Graph

Describing Individual Points or Sub-Sets of Scores WWWWe have used the distributions to describe entire sets of scores. How might we use them to describe individual scores? A rank or percentile rank is defined as the percentage of individuals in the distribution with scores at or below a particular value A percentile is used to identify an individual score’s percentile rank

Example of Frequency and Cumulative Percentage Xfcfc% % 95% 70% 30% 10%