Seminar Eight Individual ZScores and ZScore Patterns Caitlin

Seminar Eight Individual Z-Scores and Z-Score Patterns Caitlin Crawford September 20, 2007

Calculating Z-Scores • All female heights have mean 65 and standard deviation 2. 5 • All male heights have mean 70 and standard deviation 3 • Formula: z=(x-µ)/σ • In other words, the mean subtracted from your value, divided by the standard deviation

Constructing a Histogram • A histogram display values of a quantitative variable with vertical bars showing the count of values in certain interval ranges. • Steps: 1. Divide range of data into intervals of equal width 2. Find the count of observations in each interval. 3. Draw the histogram, using the horizontal axis for range of data and vertical axis for count

Commenting on Histogram • Are there large negative or positive z-scores? • What percentage of z-scores are less than 1, 2, or 3? - 68% of z-scores are between 1 and -1 - 95% of z-scores are between 2 and -2 - 99. 7% of z-scores are between 3 and -3 • Do our results conform to 68 -95 -99. 7 rule? • Would results differ for a larger class?

Example Two • How many minutes (to the nearest 10 minutes) did you spend doing homework yesterday?

Calculating Mean and Standard Deviation Steps for calculating by hand: 1. Find the mean (add up all the numbers and divide by how many there are) 2. Find the deviations from the mean 3. Find the squared deviations from the mean 4. “Average” the squared deviations, dividing their sum by the number of observations-1 5. Take the square root of the variance to find the standard deviation

Calculating Z-Scores • Use a calculator to find your individual z-score • Formula: z=(x-µ)/σ • In other words, the mean subtracted from your value, divided by the standard deviation

Constructing a Histogram • A histogram display values of a quantitative variable with vertical bars showing the count of values in certain interval ranges. • Steps: 1. Divide range of data into intervals of equal width 2. Find the count of observations in each interval. 3. Draw the histogram, using the horizontal axis for range of data and vertical axis for count

Commenting on Histogram • Are there large negative or positive z-scores? • What percentage of z-scores are less than 1, 2, or 3? - 68% of z-scores are between 1 and -1 - 95% of z-scores are between 2 and -2 - 99. 7% of z-scores are between 3 and -3 • Do our results conform to 68 -95 -99. 7 rule? • Would results differ for a larger class?

Example Three • How many hours of sleep did you get last night? -Please round to the nearest half hour

Calculating Mean and Standard Deviation Steps for calculating by hand: 1. Find the mean (add up all the numbers and divide by how many there are) 2. Find the deviations from the mean 3. Find the squared deviations from the mean 4. “Average” the squared deviations, dividing their sum by the number of observations-1 5. Take the square root of the variance to find the standard deviation

Calculating Z-Scores • Use a calculator to find your individual z-score • Formula: z=(x-µ)/σ • In other words, the mean subtracted from your value, divided by the standard deviation

Constructing a Histogram • A histogram display values of a quantitative variable with vertical bars showing the count of values in certain interval ranges. • Steps: 1. Divide range of data into intervals of equal width 2. Find the count of observations in each interval. 3. Draw the histogram, using the horizontal axis for range of data and vertical axis for count

Commenting on Histogram • Are there large negative or positive z-scores? • What percentage of z-scores are less than 1, 2, or 3? - 68% of z-scores are between 1 and -1 - 95% of z-scores are between 2 and -2 - 99. 7% of z-scores are between 3 and -3 • Do our results conform to 68 -95 -99. 7 rule? • Would results differ for a larger class?

The End Thanks for your participation!