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2.1

Describing Location in a Distribution

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Topics

Finding Percentiles

Z-Scores

Transformation of data

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Percentile

The pth percentile of a distribution is the value with p% of the observations less than or equal to it.

  • Determine the number of values less than or equal to the given value, and divide that number by the total number of values.
  • Note percentiles are measures of position, not value
    • If you see a standardized test score at the 90th percentile, it doesn’t mean you got 90% of questions right. It means that 90% of test takers scored less than or equal to your result.
  • Be careful with language - we say “at the percentile” not “in the percentile”

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Percentile

The pth percentile of a distribution is the value with p% of the observations less than or equal to it.

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The Median, Q1, and Q3 are actually Percentiles

  • Q1 is the 25th Percentile
  • The Median is the 50th Percentile
  • Q3 is the 75th Percentile

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Cumulative Relative Frequency Graph

  • Keeps a “running subtotal” of relative frequencies [or frequency] until it reaches 100% [or with frequency total sample size]
  • That means it shows percentiles

Example:

  1. Was Barack Obama, who was inaugurated at age 47 years, 169 days unusually young?

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  1. Estimate and interpret the 65th percentile of the distribution?

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Topics

Finding Percentiles

Z-Scores

Transformation of data

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Standardized Score (Z-Score)

More commonly known as Z-Scores.

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Definition: The Standardized Score (Z-Score) for an individual value in a Normal distribution tells us how many standard deviations from the mean the value falls, and in what direction.

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Formula: z =

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Standard Deviation

Value - Mean

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Standardized Score (Z-Score)

The more extreme the number (positive or negative), the farther that value is from the mean. This means that value is more unusually high (or low).

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A point’s location in the distribution depends on both distance from the center and the distribution’s spread or variation.

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Z-Scores allow us to compare values from different distributions with different Means and Standard Deviations.

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Without Z-Scores, these kinds of comparisons would be impossible to make with any statistical accuracy.

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Standardized Score (Z-Score)

On the AP Exam, you are guaranteed to have to explain what a given Z-Score means. Use the following sentence as your explanation. Write it in context:

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A Z-Score of X means:

“The given value is X Standard Deviations above/below the mean.”

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You must say it is the number of St. Dev. away from the mean

And you must give a direction, above or below.

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Standardized Score (Z-Score)

Negative Z-Scores

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Data value is less than the Mean.

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Z-Score < 0

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Described as “Below” the Mean

Z-Score = 0

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The only data value that has a Z-Score of 0 is the Mean.

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The Mean is 0 St. Dev. away from the mean.

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This is possible, however unlikely.

Positive Z-Scores

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Data value is greater than the Mean.

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Z-Score > 0

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Described as “Above” the Mean

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Standardized Score (Z-Score)

Practice

IQ scores have a mean of 100 and a standard deviation of 16. Albert Einstein reportedly had an IQ of 160.

  1. What is the difference between Einstein’s IQ and the mean?

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  1. Convert Einstein’s IQ score to a Z-Score.

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  1. If we consider “usual IQ scores” to be those that convert Z-Scores between -2 and 2, is Einstein’s IQ usual or unusual?

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  1. Interpret Albert Einstein’s Z-Score.

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Standardized Score (Z-Score)

Practice

Three students take different yet equivalent physical tests. Which student performed the best? Which student performed the worst?

Student A: Scored 144 on a test with a mean of 128 and a standard deviation of 34.

Student B Scored 90 on a test with a mean of 86 and a standard deviation of 18.

Student C: Scored 18 on a test with a mean of 15 and a standard deviation of 5.

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Standardized Score (Z-Score)

Practice

  1. Books in the library are found to have an average length of 350 pages with a standard deviation of 100 pages. How many pages does a book need to have to have a Z-Score of -2.7?

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  1. A particular leg bone for dinosaur fossils has a mean length of 5 feet. A particular bone has a length of 62 inches and a Z-Score of 0.7. What is the standard deviation of these leg bones?

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Topics

Finding Percentiles

Z-Scores

Transformation of data

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Transformations of Data

  • Sometimes it’s necessary to change our data in some uniform way.

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  • Often a result of errors or converting our data to different unit.

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  • These changes are mathematical operations done to every value in the dataset

Addition Subtraction Multiplication Division

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  • Transformations will change some combination of:

Center Position Variability (Spread)

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  • Shape will NEVER change.

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Transformations of Data

Recall from IM2 and IM3: Function Transformations

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Add or Subtract from a Function x2 +3 or (x-5)2

Effect: Shifts graph left, right, up, or down

Position Changes

Center (vertex) changes

Size of graph does NOT change

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Transformations of Data

Recall from IM2 and IM3: Function Transformations

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Multiply or Divide a Function 3x2 or (0.5x)2

Effect: Vertical or Horizontal Dilation (Stretch/Shrink)

Position Changes (wider/skinnier)

Center does NOT change (Vertex stays at origin)

Size of graph changes

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Transformations of Data

Transforming Data works very much like Function Transformations on Graphs

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Situation 1: You make an error in your data collection and you under-measured everything. To fix it you need to add 3 to every single data value in the set.

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How will this change affect measures of center, spread, and location of your data?

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The Effect of Adding or Subtracting a Constant (k) to a Dataset

  1. Add/Subtract k to each measure of Center (mean and median)
  2. Add/Subtract k to each measure of Position (5 number summary)
  3. Does not change measures of Spread (Range, IQR, and St. Dev.)

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Transformations of Data

Transforming Data works very much like Function Transformations on Graphs

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Situation 2: You collect data in feet but realize you need it in inches. To convert you need to multiply each data value by 12 to get inches.

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How will this change affect measures of center, spread, and location of your data?

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The Effect of Multiplying or Dividing a Constant (b) to a Dataset

  1. Multiply/Divide each measure of Center (mean and median) by b
  2. Multiply/Divide each measure of Position (5 number summary) by b
  3. Multiply/Divide each measure of Spread (Range, IQR, and St. Dev.) by b

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Transformations of Data

Calculating z-scores is a Transformation.

You take each data value and subtract the mean and then divide by the St. Dev.

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When you are performing multiple transformations, follow Order of Operations.

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Transformations of Data

Practice

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HW 2.1

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HW 2.1

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HW 2.1

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HW 2.1

p. 104-108 #11, 13, 19, 23, 27 and 33

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HW 2.1

p. 104-108 #11, 13, 19, 23, 27 and 33

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HW 2.1

p. 104-108 #11, 13, 19, 23, 27 and 33

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HW 2.1

p. 104-108 #11, 13, 19, 23, 27 and 33

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