Today's Objectives: FIND and INTERPRET the standardized score (z-score) of an individual value within a distribution of data. DESCRIBE the effect of adding, subtracting, multiplying by, or dividing by a constant on the shape, center, and spread of a distribution of data.
Just like we used median and IQR to find the percentile (placement) of data, we also have a way to compare a specific piece of data in comparison to the whole when we are using mean and standard deviation (so we have an approximately normal distribution). We do this by STANDARDIZING our distributions. This puts the distributions into the same form, which allows us to make comparisons between them. It also allows us to find percentages/percentiles at any data point under the curve, although that's not today's lesson.
A z-score tells us how many standard deviations from the mean an observation falls, and in what direction. If x is an observation from a distribution that has known mean and standard deviation, the standardized score of x is: point data To Loading... where x is the data point we want information about. mean T standard deviation
Any value of x from our original distribution, minus the mean and divided by the standard deviation gives you the z-score. Although the formula uses notation for the population mean and standard deviation, we can do the same procedure with the sample mean and standard deviation. Positive z-scores are to the right of the mean. Negative z-scores are to the left of the mean. Z-scores are equal to the number of standard deviations you are from the mean in a standard normal distribution. * very Important!
5 Example: In 2012, the mean number of wins for teams in Major League Baseball was 81, o with a standard deviation of 11.9 0 wins. Find and interpret the z-scores for the following teams. µ= 81, 0=11.9 a) The New York Yankees, with 95 wins. z= 95-81 Tg = 1.18 Loading... The Yankees number of wins is 1.18 standard deviations above the mean for all Major League baseball teams. b) The New York Mets with 75 wins. Z = 75-81 # = -. 504 ux The Mets ' number wins is. of below the baseball teams. g standard deviations average number of wins for all Major League
. AP Stats: 2B ~ Z-scores & transformations Example: Compute the standardized score for each performance using the information in the table. Which player had the most outstanding performance relative to his peers? Year Player HR Mean SD 1927 Babe Ruth 60 7.2 9.7 1961 Roger Maris 61 18.8 13.4 1998 Mark McGwire 70 20.7 12.7 2001 Barry Bonds 73 21.4 13.2 zenif s*fiotie riyf F ei *f BE a, 13.2 Babe Ruth was mud farther above average In comparison to his peers He is definitely the greatest!
Transforming data: We transformed our data by standardizing the values (finding the z- score), but there are other transformations that can affect our distributions. A transformation changes our original units of measure to a different scale. What kind of common transformations can you think of?
Add. The Effect of Adding (or subtracting) a Constant: Adding the same number a to (subtracting a from) each observation: Adds a to (or subtracts a from) measures of center and location (mean, median, quartiles, percentiles), but Does not change the shape of the distribution or measures of spread (range, IQR, standard deviation). Why? If we add the same value to every data piece, it just slides the distribution along the *axis, it does not change the variation itself in any way For example : Say the range of a distribution Is from 10 to 20, distance of 10. 32 to everything, the range goes from or a 42 to 52, or a distance of 10.
The Effect of Multiplying (or dividing) a Constant: Multiplying (or dividing) each observation by the same number b: Multiplies (divides) measures of center and location (mean, median, quartiles, percentiles) by b. Multiplies (divides) measures of spread (range, IQR, standard deviation) by b, but Does not change the shape of the distribution.
Example: In 2010, taxicabs in New York City charged an initial fee of $2.50 plus $2 per mile. At the end of the month, a businessman collects all of his taxicab receipts and finds that the mean fare he paid was $15.45, with a standard deviation of $10.20. A) What equation describes the transformation? Let X= # miles Fare =2.50 t ZX B) What are the mean and standard deviation of the lengths of his cab rides in MILES? add multiply and Fare -2.50=2 X Imuestxrzare -2251 = 15.45ft = tempted - X= Fare 2.50 = I on,,e, = srzart.ly#=5m.@ Divide only
EXAMPLE: Maria measures the lengths of 5 cockroaches that she found at school. Here are her results in inches: 1.4 2.2 1.1 1.6 1.2 A) Find the mean and standard deviation: Using tvarstats : I= 1.5 s =. 436 Loading... B) Maria s teacher wants the results in centimeters, not inches. Convert the mean and standard deviation to centimeters. (Remember 2.54 cm = 1 in) Then cm=lin 2.54 XTm=z g= -591cm m=g4.}6y- =. 172cm
Assignment: page 101: #13-24, 26-28 Read pp. 103-112