Edpsy 511 Homework 1: Due 2/6

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: Categorical data: obtained by determining the frequency of occurrences in each of several categories 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 Summary Measures Central Tendency Quartile Variation Arithmetic Mean Median Mode Range Variance Standard Deviation

Measures of Central Tendency Average (Mean) Median Mode

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) 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14 Mean = 5 Mean = 6

Mean of Grouped Frequency X f fX 10 1 9 3 8 2 7 4 6 5 Total N 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.

Group xbar N f(xbar) 1 30 10 2 25 15 3 40

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 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14 Median = 5 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 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 No Mode Mode = 9

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, s2, 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, s2, is the sample variance, and is used to estimate the actual population variance, s 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 (Definitional formula) Raw Score Mean X – X (X – X)2 85 54 31 961 80 54 26 676 70 54 16 256 60 54 6 36 55 54 1 1 50 54 -4 16 45 54 -9 81 40 54 -14 196 30 54 -24 576 25 54 -29 841 Variance (SD2) = Σ(X – X)2 N-1 3640 9 = =404.44 √ Standard deviation (SD) = Σ(X – X)2 N-1

Definitional vs. Computational An equation that defines a measure Computational An equation that simplifies the calculation of the measure

Calculate the variance using the computational and definitional formulas. 10, 12, 12, 12, 13, 13, 14

In small groups: calculate the variance using the computational and definitional formulas. 2, 8, 9, 11, 15, 17, 20

Comparing Standard Deviations Data A Mean = 15.5 S = 3.338 11 12 13 14 15 16 17 18 19 20 21 Data B Mean = 15.5 S = .9258 11 12 13 14 15 16 17 18 19 20 21 Data C Mean = 15.5 S = 4.57 11 12 13 14 15 16 17 18 19 20 21

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 68-95-99.7 rule

The Normal Curve

Different Distributions Compared

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 10 + 50).

Probability Areas Between the Mean and Different Z Scores

Examples of Standard Scores