Normal Distribution Symmetrical Continuous Unimodal Mean= Median = Mode Asymptotic Tail Almost 100% of scores fits between -3 and +3 stdev from the mean

How to Determine if Nearly Normal Unimodal Relatively Symmetric

Standard Scores: Z score + z score = right of the mean z score = left of the mean Can compare z scores from different distributions

z-score looks at one data point, x, and tells us how many standard deviations x is from the mean. You need to know this for the quiz Z =±1 68% of all data points are within 1 stdev.of the mean Z = ±2 95% are within 2 stdev. of the mean Z = ±3 99.7% are within 3 stdev. of the mean

100 m Dash Shot Put Long Jump Ashton Eaton 10 sec 66’ 26’ Trey Hardee 9.8 sec 60’ 26.25’ Leonel Suarez 10.6 sec 63’ 27.5’ Z = 0 Z= 1 Z= -3 Z = 2 Z= 0 Z= 1 Z = 0 Z= 0.5 Z= 3 Let’s talk about scoring the decathlon. Silly example, but suppose two competitors tie in each of the first eight events. In the ninth event, the high jump, one clears the bar 1 in. higher. Then in the 1500-meter run the other one runs 5 seconds faster. Who wins? It boils down to knowing whether it is harder to jump an inch higher or run 5 seconds faster. We have to be able to compare two fundamentally different activities involving different units. Standard deviations to the rescue! If we knew the mean performance (by world-class athletes) in each event, and the standard deviation, we could compute how far each performance was from the mean in SD units (called z-scores). So consider the three athletes’ performances shown below in a three event competition. Note that each placed first, second, and third in an event. Who gets the gold medal? Who turned in the most remarkable performance of the competition? Mean 10 sec 60’ 26’ St Dev .2 sec 3’ 0.5’

How to use the z-score Watch & Listen…

Handout #1-10 Example For a given test, what is the probability that a score will be 110 or above? mean = 100 standard deviation = 10 16%

Probability only works when z score is ±1, ±2, or ±3 Two ways to computer : 1) 68 – 95 – 97.5 rule only works when z score is ±1, ±2, or ±3 2) z-scores and z-table: A- 56 & A-57

Finding probability using z-table #1: find the z score #2: look up the probability on the z-table (link on website in tools) Separate your score after 10th place: z = 2.33 First two numbers: 2.3 tells how far down the table Second two numbers: 3 tells how far over

Normal curve for test scores What is the probability of getting 70% or lower on a test if the z-score for 70 is 1.5? 70 z-score = 1.5 Normal curve for test scores 93.319% Need to find the percent that represents the grey area. This percent equals the probability of getting 70% or less

How to find a basic prob IQ scores have the distribution: N(100,16) What is the probability of having a score less than 125? #1 draw a picture #2 write a probability statement #3 find the z-score #4 look up the probability

Finding prob btwn two points EXAMPLE: What percent of scores are between 110 and 125 where mean= 100 and stdev = 10 15.379% 100 110 125 #1: compute z score for 110: z = 1 #2: compute z score for 125: z = 2.5 #3: Find tHE PROBABILITY OF EACH (Z-TABLE) P(110) = 84% p(125) = 99.379% #4: subtract: 99.379% - 84% = 15.379%