Comparisons across normal distributions Z -Scores.

Comparisons across normal distributions Z -Scores

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

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

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 VO 2max vs. Vertical Jump BMI vs. %BodyFat vs. Waist Circumference

Minimal Statistics 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) Describe the typical score, the spread of scores, and the number of cases

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

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

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

Here is our normal distribution example from last week X = 70 SD = 10 7060805090 40100 2.3% 34.1% 13.6% -3-20123Z = If a subject scored 70, their z-score would be 0

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

Here is our normal distribution example from last week X = 70 SD = 10 7060805090 40100 2.3% 34.1% 13.6% -3-20123Z = What is the z-score of a subject that got: 80? 50? 100?

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

Here is our normal distribution example from last week X = 70 SD = 10 7060805090 40100 2.3% 34.1% 13.6% -3-20123Z = 34% of scores still fall between 0 and 1 z-score

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

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

What upper and lower limits include 95% of BMI scores? If one boys BMI is 22 kg/m 2 and anothers WC is 70 cm, which of the two has the highest adiposity? Example: Mean BMI and WC in elementary school boys

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

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)

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)

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

Numerator Monster How much do you pay in car insurance every 6 months? Sohow much is State Farm keeping?

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 its 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?)

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

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

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)

Frequency Stem & Leaf 2.00 Extremes (=

Comparisons across normal distributions Z -Scores.

Documents

Transcript of Comparisons across normal distributions Z -Scores.