Edpsy 511 Exploratory Data Analysis Homework 1: Due 9/19

Shapes of Distributions ► Normal distribution ► Positive Skew  Or right skewed ► Negative Skew  Or left skewed

How is this variable distributed?

Descriptive Statistics

Statistics vs. Parameters ► A parameter is a characteristic of a population.  It is a numerical or graphic way to summarize data obtained from the population ► A statistic is a characteristic of a sample.  It is a numerical or graphic way to summarize data obtained from a sample

Types of Numerical Data ► There are two fundamental types of numerical data: 1) Categorical data: obtained by determining the frequency of occurrences in each of several categories 2) Quantitative data: obtained by determining placement on a scale that indicates amount or degree

Techniques for Summarizing Quantitative Data ► Frequency Distributions ► Histograms ► Stem and Leaf Plots ► Distribution curves ► Averages ► Variability

Summary Measures Central Tendency Arithmetic Mean Median Mode Quartile Summary Measures Variation Variance Standard Deviation Range

Measures of Central Tendency Central Tendency Average (Mean)MedianMode

Mean (Arithmetic Mean) ► Mean (arithmetic mean) of data values  Sample mean  Population mean Sample Size Population Size

Mean ► The most common measure of central tendency ► Affected by extreme values (outliers) Mean = 5Mean = 6

Mean of Grouped Frequency XffX TotalN 21

Weighted Mean A form of mean obtained from groups of data in which the different sizes of the groups are accounted for or weighted.

GroupxbarNf(xbar)

Median ► Robust measure of central tendency ► Not affected by extreme values ► In an Ordered array, median is the “middle” number  If n or N is odd, median is the middle number  If n or N is even, median is the average of the two middle numbers Median = 5

Mode ► A measure of central tendency ► Value that occurs most often ► Not affected by extreme values ► Used for either numerical or categorical data ► There may may be no mode ► There may be several modes Mode = No Mode

The Normal Curve

Different Distributions Compared

Variability ► Refers to the extent to which the scores on a quantitative variable in a distribution are spread out. ► The range represents the difference between the highest and lowest scores in a distribution. ► A five number summary reports the lowest, the first quartile, the median, the third quartile, and highest score.  Five number summaries are often portrayed graphically by the use of box plots.

Variance ► The Variance, s 2, represents the amount of variability of the data relative to their mean ► As shown below, the variance is the “average” of the squared deviations of the observations about their mean ► The Variance, s 2, is the sample variance, and is used to estimate the actual population variance,  2

Standard Deviation ► Considered the most useful index of variability. ► It is a single number that represents the spread of a distribution. ► If a distribution is normal, then the mean plus or minus 3 SD will encompass about 99% of all scores in the distribution.

Calculation of the Variance and Standard Deviation of a Distribution √ Raw ScoreMeanX – X(X – X) Variance (SD 2 ) = Σ(X – X) 2 N-1 = = Standard deviation (SD) = Σ(X – X) 2 N-1

Comparing Standard Deviations Mean = 15.5 S = Data B Data A Mean = 15.5 S = Mean = 15.5 S = 4.57 Data C

Facts about the Normal Distribution ► 50% of all the observations fall on each side of the mean. ► 68% of scores fall within 1 SD of the mean in a normal distribution. ► 27% of the observations fall between 1 and 2 SD from the mean. ► 99.7% of all scores fall within 3 SD of the mean. ► This is often referred to as the rule

Fifty Percent of All Scores in a Normal Curve Fall on Each Side of the Mean

Probabilities Under the Normal Curve

Standard Scores ► Standard scores use a common scale to indicate how an individual compares to other individuals in a group. ► The simplest form of a standard score is a Z score. ► A Z score expresses how far a raw score is from the mean in standard deviation units. ► Standard scores provide a better basis for comparing performance on different measures than do raw scores. ► A Probability is a percent stated in decimal form and refers to the likelihood of an event occurring. ► T scores are z scores expressed in a different form (z score x ).

Probability Areas Between the Mean and Different Z Scores

Examples of Standard Scores