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  1. Standard Scores Comparisons across normal distributions Z-Scores

  2. Overview • Plan for the night • Z-scores • Definition • Calculation • Use • Graphing Data/Distributions • Frequencies/Percentages • Charts/Graphs

  3. Last time… • Last week we covered • Measures of Central Tendency • Mean, Mode, Median • Measures of Variability • Range, 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/datapoint

  4. Is 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. GRE • 10-yd dash time vs. 40-yd dash time • Free-throw% vs. FG% vs. 3-Point% • Example (w/different unit of measure): • ERA vs. WHIP • VO2max vs. Vertical Jump • BMI vs. %BodyFat vs. Waist Circumference

  5. Minimal Statistics Describe the “typical” score, the “spread” of scores, and the number of cases • Mean • SD • m Z-scores • Combine 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 SD • Examples: • z-score = 1.00 (1 SD above mean) • z-score = -2.00 (2 SD below mean)

  6. Z-score = 1.0: GRAPHICALLY 84% of scores smaller than this Z = 1 Recall – 50% of scores are below the mean + 34% of scores between the mean and 1 SD above

  7. Calculating z-scores OR Calculate Z for each of the following situations:

  8. Other features of z-scores • 1) The Mean of a distribution of z-scores = 0 • Recall the mean is the balance point of a distribution, where deviation scores sum to 0 • A z-score of 0 is equivalent to scoring the mean

  9. Here is our normal distribution example from last week X = 70 SD = 10 If a subject scored 70, their z-score would be 0 34.1% 34.1% 13.6% 13.6% 2.3% 2.3% Z = -3 40 50 -2 -1 60 70 0 80 1 2 90 100 3

  10. Other features of z-scores • 1) The Mean of a distribution of z-scores = 0 • Recall the mean is the balance point of a distribution, where deviation scores sum to 0 • A z-score of 0 is equivalent to scoring the mean • 2) The SD of a distribution of z-scores = 1 • Since SD is unit of measurement, when the mean is z=0 then the mean + 1 SD = a z-score of 1

  11. Here is our normal distribution example from last week What is the z-score of a subject that got: 80? 50? 100? X = 70 SD = 10 34.1% 34.1% 13.6% 13.6% 2.3% 2.3% Z = -3 40 -2 50 -1 60 0 70 80 1 2 90 100 3

  12. Other features of z-scores • 1) The Mean of a distribution of z-scores = 0 • Recall the mean is the balance point of a distribution, where deviation scores sum to 0 • A z-score of 0 is equivalent to scoring the mean • 2) The SD of a distribution of z-scores = 1 • Since 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 distribution • Even though you are changing the unit of measurement, this does not change the “look” of the distribution when plotted

  13. Here is our normal distribution example from last week 34% of scores still fall between 0 and 1 z-score X = 70 SD = 10 34.1% 34.1% 13.6% 13.6% 2.3% 2.3% Z = -3 40 -2 50 -1 60 0 70 80 1 2 90 100 3

  14. Z-score Comparison • As stated, z-scores standardize different distributions allowing you to make comparisons regardless of the unit of measure • Bart’s score • SAT Exam 450 (mean 500, SD 100) • Lisa’s score • ACT Exam 24 (mean 18, SD 6) • Who scored higher? Bart: (450 – 500)/100 = - 0.5 Lisa: (24 – 18)/6 = 1

  15. Z-scores & the normal curve • For any z-score, we can calculate the percentage of scores between it and the mean; all scores below it & all above it • Tons of online calculators: • http://www.measuringusability.com/normal_curve.php

  16. Example: Mean BMI and WC in elementary school boys What upper and lower limits include 95% of BMI scores? If one boy’s BMI is 22 kg/m2 and another’s WC is 70 cm, which of the two has the highest adiposity?

  17. FREQUENCY DISTRIBUTIONS & GRAPHING

  18. Nomenclature/Terminology • Frequency: number of cases or subjects or occurrences in a distribution • Represented with f • i.e. f = 12 for a score of 25 • 12 occurrences of 25 in the sample

  19. Nomenclature/Terminology • Percentage: Number of cases or subjects or occurrences expressed per 100 • Represented with P or % • Ex. f=12 for a score of 25 when n=25 • P = 12/25*100 = 48% (of scores were 25)

  20. Warning • Should report the f when presenting percentages • i.e. 80% of the elementary students came from a family with an income < $25,000 • different interpretation if n=5 compared to n=100 • Reported in literature as • f = 4 (80%) OR • 80% (f = 4) OR • 80% (n = 4)

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

  22. Numerator Monster How much do you pay in car insurance every 6 months? So…how much is State Farm keeping?

  23. Frequency Distributions • Graphically 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 it’s always easier to visually examine it • Check 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?)

  24. Frequency Distribution of Math Test Scores: SPSS Output • 40 items on exam • Most students >34 • skewed (more scores at one end of the scale)

  25. Cumulative frequencies &, Cumulative percentages • Cumulative Percentage: how many subjects atand below a given score? • i.e., 33.3% of students scored a 32 or lower

  26. Eyeball check of data: Intro to (brute force) graphing with SPSS • Stem and Leaf Plot: quick viewing of data distribution • Boxplot: visual representation of many of the descriptive statistics discussed last week • Bar Chart: frequency of all cases • Histogram: malleable bar chart • Scatterplot: displays all cases based on two values of interest (X & Y) • Note: compare to our previous discussion of distributions (normal, positively skewed, etc…)

  27. Stem and Leaf Plots • Fast look at shape of distribution • shows f numerically & graphically • stem is value, leaf is f Frequency Stem & Leaf 2.00 Extremes (=<25.0) 2.00 28 . 00 2.00 29 . 00 1.00 30 . 0 1.00 31 . 0 3.00 32 . 000 1.00 33 . 0 6.00 34 . 000000 3.00 35 . 000 4.00 36 . 0000 8.00 37 . 00000000 Stem width: 1 Each leaf: 1 case

  28. Stem and Leaf Plots • Another way of doing a stemplot • Babe Ruth’s home runs in each of 14 seasons with the NY Yankees • 54, 59, 35, 41, 46, 25, 47, 60, 54, 46, 49, 46, 41, 34, 22 2 25 3 45 4 1166679 5 449 6 0

  29. Stem and Leaf Plots • Back-to-back stem plots allow you to visualize two data sets at the same time • Babe Ruth vs. Roger Maris MarisRuth 0 1 2 25 3 45 4 1166679 5 449 6 0 8 643 863 93 1

  30. Boxplots Maximum Q3 Median Q1 Minimum Note: we can also do side-by-side boxplots for a visual comparison of data sets

  31. Format of Bar Chart Y axis f X axis scores/categories

  32. Test score data as Bar Chart

  33. Format of Histogram (similar to Bar) Y axis f X axis Can be manipulated scores/categories

  34. Test score data as Histogram

  35. Test score data as Revised Histogram

  36. Scatterplot • Quick way to visualize the data & see trends, patterns, etc… • This plot visually shows the relationship between BMI and WC in a group of elementary school boys

  37. Scatterplot Somebody shook their pedometer for 2 hours a day… • Here’s the relationship between females Steps/day and waist circumference

  38. Scatterplot • Outlier removed • This will impact any statistical tests you run (correlations, regression, etc…)

  39. Take home message • Z-scores: • A simple combination of Mean and SD • Allow comparisons regardless of unit of measurement • Always plot your data first! • Descriptive statistics (like Mean/SD) are generally presented along with graphical representations of the distribution • A histogram (for single variable) and scatterplot (for paired variables) are most commonly used • Check for outliers! Is the value plausible?

  40. Upcoming… • Homework = Cronk 3.5 & all of Chapter 4 • Blackboard description upcoming • We will examine relationships between variables next week • Think about those scatterplots…do statistical relationships exist between those variables? How strong? In what direction? • In-class activity 3…