FREQUENCY DISTRIBUTIONS & GRAPHING

Standard ScoresComparisons across normal distributions Z-ScoresOverviewPlan for the nightZ-scoresDefinitionCalculationUse

Graphing Data/DistributionsFrequencies/PercentagesCharts/GraphsLast timeLast week we coveredMeasures of Central TendencyMean, Mode, MedianMeasures of VariabilityRange, IQR, SIQR, Standard Deviation

The most commonly used of the above are Mean (SD)These two measures can be combined to further describe the position of a score/datapointIs that a good score?Mean and SD are useful, but sometimes we need to make comparisons between different measures

Example (w/ same units of measure): SAT vs. ACT vs. GRE10-yd dash time vs. 40-yd dash timeFree-throw% vs. FG% vs. 3-Point%

Example (w/different unit of measure):ERA vs. WHIPVO2max vs. Vertical JumpBMI vs. %BodyFat vs. Waist Circumference

Minimal StatisticsMeanSDm

Z-scoresCombine the mean w/ SD to create a new unit of measurement (Standardizes Scores)Clearly identifies a score as above or below the mean AND expresses a score in units of SDExamples: z-score = 1.00 (1 SD above mean)z-score = -2.00 (2 SD below mean)Describe the typical score, the spread of scores, and the number of casesZ-score = 1.0: GRAPHICALLY

Z = 184% of scores smaller than thisRecall 50% of scores are below the mean + 34% of scores between the mean and 1 SD aboveCalculating z-scores

Calculate Z for each of the following situations:

OR

Other features of z-scores1) The Mean of a distribution of z-scores = 0 Recall the mean is the balance point of a distribution, where deviation scores sum to 0A z-score of 0 is equivalent to scoring the meanHere is our normal distribution example from last weekX = 70SD = 107060805090401002.3%2.3%34.1%34.1%13.6%13.6%-3-2-10123Z =If a subject scored 70, their z-score would be 0Other features of z-scores1) The Mean of a distribution of z-scores = 0 Recall the mean is the balance point of a distribution, where deviation scores sum to 0A z-score of 0 is equivalent to scoring the mean

2) The SD of a distribution of z-scores = 1Since SD is unit of measurement, when the mean is z=0 then the mean + 1 SD = a z-score of 1

Here is our normal distribution example from last weekX = 70SD = 107060805090401002.3%2.3%34.1%34.1%13.6%13.6%-3-2-10123Z =What is the z-score of a subject that got: 80? 50? 100?Other features of z-scores1) The Mean of a distribution of z-scores = 0 Recall the mean is the balance point of a distribution, where deviation scores sum to 0A z-score of 0 is equivalent to scoring the mean

2) The SD of a distribution of z-scores = 1Since SD is unit of measurement, when the mean is z=0 then the mean + 1 SD = a z-score of 1

3) A z-score distribution is same shape as raw score distributionEven though you are changing the unit of measurement, this does not change the look of the distribution when plottedHere is our normal distribution example from last weekX = 70SD = 107060805090401002.3%2.3%34.1%34.1%13.6%13.6%-3-2-10123Z =34% of scores still fall between 0 and 1 z-scoreZ-score ComparisonAs stated, z-scores standardize different distributions allowing you to make comparisons regardless of the unit of measureBarts scoreSAT Exam 450 (mean 500, SD 100)Lisas scoreACT Exam 24 (mean 18, SD 6)

Who scored higher?Bart: (450 500)/100 = - 0.5Lisa: (24 18)/6 = 1Z-scores & the normal curveFor any z-score, we can calculate the percentage of scores between it and the mean; all scores below it & all above itTons of online calculators:http://www.measuringusability.com/normal_curve.phpWhat upper and lower limits include 95% of BMI scores?

If one boys BMI is 22 kg/m2 and anothers WC is 70 cm, which of the two has the highest adiposity?Example: Mean BMI and WC in elementary school boys

FREQUENCY DISTRIBUTIONS & GRAPHINGNomenclature/TerminologyFrequency: number of cases or subjects or occurrences in a distribution

Represented with f

i.e. f = 12 for a score of 2512 occurrences of 25 in the sampleNomenclature/TerminologyPercentage: Number of cases or subjects or occurrences expressed per 100

Represented with P or %

Ex. f=12 for a score of 25 when n=25P = 12/25*100 = 48% (of scores were 25)WarningShould report the f when presenting percentagesi.e. 80% of the elementary students came from a family with an income < $25,000different interpretation if n=5 compared to n=100

Reported in literature asf = 4 (80%) OR80% (f = 4) OR80% (n = 4)Numerator Monster

Pantagraph, 6/13/00Pantagraph reported that State Farm paid out over 1 Billion in dividends to customers in the United StatesNumerator Monster

How much do you pay in car insurance every 6 months?Sohow much is State Farm keeping?Frequency DistributionsGraphically displaying the data should ALWAYS come before any type of statistical analysis

Measures of central tendency and variability will give you a feeling for the distribution of the data but its always easier to visually examine itCheck for normality (are data normally distributed?)Check for outliers (are any subjects sticking out as odd?)Check of potential associations (might two variables relate to each other?)Frequency Distribution of Math Test Scores: SPSS Output40 items on examMost students >34skewed (more scores at one end of the scale)

Cumulative frequencies &, Cumulative percentagesCumulative Percentage: how many subjects at and below a given score?

i.e., 33.3% of students scored a 32 or lowerEyeball check of data: Intro to (brute force) graphing with SPSSStem and Leaf Plot: quick viewing of data distributionBoxplot: visual representation of many of the descriptive statistics discussed last weekBar Chart: frequency of all casesHistogram: malleable bar chartScatterplot: displays all cases based on two values of interest (X & Y)Note: compare to our previous discussion of distributions (normal, positively skewed, etc) Frequency Stem & Leaf 2.00 Extremes (=