1 Describing Location in a Distribution
2 Activity: Where do I stand? (p. 84)In this activity, you and your classmates will explore ways to describe where you stand within a distribution Figure out your height in inches. Round to the nearest inch Mark your height on the dotplot Calculate what percent of students in the class have a height less than yours. This is your percentile in the distribution of heights Work with a partner (if you want) to calculate the mean and standard deviation of the class’s heights. Where does your height fall in relation to the mean? Above or below? How far above or below? How many standard deviations above or below the mean is it? This last number is the z-score for your height
3 Percentile The Pth percentile of a distribution is the value in the with p percent of the observations less than it So, if 25% of the teachers did worse than Mr. Wetherbee on their Praxis scores, at what percentile was his score?
4 Percentile The Pth percentile of a distribution is the value in the with p percent of the observations less than it So, if 25% of the teachers did worse than Mr. Wetherbee on their Praxis scores, at what percentile was his score? 25th
5 Think about it Why are percentiles useful?
6 Think about it Why are percentiles useful?Show us information about how an observation/individual compares to others Does this by showing what percent are below that individual Often used to report scores on standardized tests
7 Let’s Try It The following data represents the number of wins of each of the 30 major league baseball team in 2009, ranked from fewest to most: 59,62,64,65,65,70,70,74,75,75,75,78,79,80,83,84,85,86, 86,87,87,87,88,91,92,93,95,95,97,103 The Rockies won 92 games. At what percentile do they fall? The Yankees won 103 games. At what percentile do they fall?
8 Let’s Try It The following data represents the number of wins of each of the 30 major league baseball team in 2009, ranked from fewest to most: 59,62,64,65,65,70,70,74,75,75,75,78,79,80,83,84,85,86,86,87 ,87,87,88,91,92,93,95,95,97,103 The Rockies won 92 games. At what percentile do they fall? 24 of 30 (.8) teams did worse than the Rockies So they are at the 80th percentile The Yankees won 103 games. At what percentile do they fall? 29 of 30 teams did worse than the Yankees, so they are at about the 97th percentile
9 Z-score The Z-score for a value in a dataset is the number of standard deviations that it falls away from the mean of the dataset 𝑧= 𝑥− 𝑥 𝑠 𝑥 Or, in natural language, the value minus the mean all divided by the standard deviation
10 Why are z-scores useful?
11 Why are z-scores useful?Also gives us information about how an individual compares to others Does this by showing how many standard deviations above or below the mean an individual is This is particularly useful when comparing dissimilar things Questions like: “Am I better at math than my friend is at English?” “Is the Broncos’ defense better at defense than the Steelers’ offense is at offense?”
12 Is the Broncos’ defense better than the Steelers’ offense? (2015-2016)Let’s calculate it Mean of offensive yards/play: 5.45 Standard deviation of yards/play: .3297 Steelers’ offense: 6.22 yards/play ( )/.3297= 2.336= z-score Steelers’ offense is standard deviations above the mean Mean of defensive yards/play: 5.45 Standard deviation of yards/play: .4235 Broncos’ defense: 4.48 yards/play ( )/.4235= But remember, on defense a lower number is better However, the Steelers’ offense does seem to be just a little bit better than the Broncos’ defense At least in terms of yards/play
13 Cumulative Relative FrequencyWhat does this graph tell us?
14 Cumulative Relative FrequencyWhat does this graph tell us? President Obama was 47 when inaugurated. Was he abnormally young?
15 Cumulative Relative FrequencyWhat does this graph tell us? President Obama was 47 when inaugurated. Was he abnormally young? Roughly the 11th percentile—89% were older than him.
16 What happens when we transform our data?Adding or subtracting a number to each observation How does it affect shape: Outliers: Center: Spread:
17 What happens when we transform our data?Adding or subtracting a number (X) to each observation How does it affect shape: no effect Center: shifts it by X Spread: no effect
18 What happens when we transform our data?Multiplying by a number (X) to each observation How does it affect shape: Center: Spread:
19 What happens when we transform our data?Multiplying by a number (X) to each observation How does it affect shape: no effect Center: multiplies it by X Spread: multiplies it by X
20 What happens when we transform our data?What happens to our Z-scores when we transform our data?
21 What happens when we transform our data?What happens to our Z-scores when we transform our data? Does not change! This allows us to compare Z-scores of variables that are on many different scales Because the z-score is telling us in terms of standard deviations
22 Transforming Data The average (mean) line at an ice cream shop is 8 minutes, with a standard deviation of 2 minutes The shop is small and only has one employee. The employee quits, and the shop hires an new employee. Because he is new, he takes twice as long to get everyone’s ice cream What happens to the distribution of wait time? Mean? Standard deviation?
23 Transforming Data The average (mean) line at an ice cream shop is 8 minutes, with a standard deviation of 2 minutes The shop is small and only has one employee. The employee quits, and the shop hires an new employee, named Jim. Because Jim is new, he takes twice as long to get everyone’s ice cream What happens to the distribution of wait time? Mean? Now 16 minutes Standard deviation? Now 4 minutes
24 Transforming Data Jim never gets faster, so the shop fires him, and hires Ashley Ashley is able to improve upon Jim’s times by 2 minutes What happens to the distribution of wait times? Mean? Standard deviation?
25 Transforming Data Jim never gets faster, so the shop fires him, and hires Ashley Ashley is able to improve upon Jim’s times by 2 minutes What happens to the distribution of wait times? Mean? 14 minutes Standard deviation? 4 minutes
26 An Example At what percentile is Ireland in this distribution?Country/ Territory GDP per Capita Jordan $11,100 Bermuda $85,700 Marshall Islands $3,300 Congo $6,800 Brazil $14,800 Greece $26,800 Ireland $69,400 Hong Kong $58,100 Solomon Islands $2,000 Laos $5,700 At what percentile is Ireland in this distribution? Find the z-score for Brazil in this distribution Create a cumulative relative frequency graph for this distribution Using your cumulative relative frequency graph, at about what percentile is the United States ($57,300)
27 80th percentile Z-score: -.438 See Below 69th Percentile (ish) sidenote: in reality 91st