3 Statistics vs. Parameters A parameter is a characteristic of a population.It is a numerical or graphic way to summarize data obtained from the populationA statistic is a characteristic of a sample.It is a numerical or graphic way to summarize data obtained from a sample
4 Types of Numerical Data There are two fundamental types of numerical data:Categorical data: obtained by determining the frequency of occurrences in each of several categoriesQuantitative data: obtained by determining placement on a scale that indicates amount or degree
5 Techniques for Summarizing Quantitative Data Frequency DistributionsHistogramsStem and Leaf PlotsDistribution curvesAveragesVariability
13 Median Robust measure of central tendency Not affected by extreme valuesIn an Ordered array, median is the “middle” numberIf n or N is odd, median is the middle numberIf n or N is even, median is the average of the two middle numbersMedian = 5Median = 5
14 Mode A measure of central tendency Value that occurs most often Not affected by extreme valuesUsed for either numerical or categorical dataThere may may be no modeThere may be several modesNo ModeMode = 9
15 VariabilityRefers 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.
16 VarianceThe Variance, s2, represents the amount of variability of the data relative to their meanAs shown below, the variance is the “average” of the squared deviations of the observations about their meanThe Variance, s2, is the sample variance, and is used to estimate the actual population variance, s 2
17 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.
18 Calculation of the Variance and Standard Deviation of a Distribution (Definitional formula) RawScore Mean X – X (X – X)2Variance (SD2) =Σ(X – X)2N-136409==404.44√Standard deviation (SD) =Σ(X – X)2N-1
19 Definitional vs. Computational An equation that defines a measureComputationalAn equation that simplifies the calculation of the measure
20 Calculate the variance using the computational and definitional formulas. 10, 12, 12, 12, 13, 13, 14
21 In small groups: calculate the variance using the computational and definitional formulas. 2, 8, 9, 11, 15, 17, 20
22 Comparing Standard Deviations Data AMean = 15.5S = 3.338Data BMean = 15.5S = .9258Data CMean = 15.5S = 4.57
23 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
28 Standard ScoresStandard 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 ).
29 Probability Areas Between the Mean and Different Z Scores