Measures of Relative Standing and Density Curves

1
Lesson 2-1

• Measures of Relative Standing and Density Curves

2
Knowledge Objectives

• Explain what is meant by a standardized value
• Define Chebyshevs inequality, and give an
  example of its use
• Explain what is meant by a mathematical model
• Define a density curve
• Explain where the median and mean of a density
  curve can be found

3
Construction Objectives

• Compute the z-score of an observation given the
  mean and standard deviation of a distribution
• Compute the pth percentile of an observation
• Describe the relative position of the mean and
  median in a symmetric density curve and in a
  skewed density curve

4
Vocabulary

• Density Curve the curve that represents the
  proportions of the observations and describes
  the overall pattern
• Mathematical Model an idealized representation
• Median of a Density Curve is the equal-areas
  point and denoted by M or Med
• Mean of a Density Curve is the balance point
  and denoted by ? (Greek letter mu)
• Normal Curve a special symmetric, mound shaped
  density curve with special characteristics

5
Vocabulary

• Pth Percentile the observation that in rank
  order is the pth percentile of the sample
• Standard Deviation of a Density Curve is
  denoted by ? (Greek letter sigma)
• Standardized Value a z-score
• Standardizing converting data from original
  values to standard deviation units
• Uniform Distribution a symmetric rectangular
  shaped density distribution

6
Sample Data

• Consider the following test scores for a small
  class

79 81 80 77 73 83 74 93 78 80 75 67 73
77 83 86 90 79 85 83 89 84 82 77 72
Jennys score is noted in red. How did she
perform on this test relative to her peers?
6 7 7 2334 7 5777899 8 00123334 8
569 9 03
6 7 7 2334 7 5777899 8 00123334 8
569 9 03
7
Standardized Value

• One way to describe relative position in a data
  set is to tell how many standard deviations above
  or below the mean the observation is.

8
Calculating z-scores

• Consider the test data and Julias score.

79 81 80 77 73 83 74 93 78 80 75 67 73
77 83 86 90 79 85 83 89 84 82 77 72
According to Minitab, the mean test score was 80
while the standard deviation was 6.07 points.
Julias score was above average. Her
standardized z-score is
Julias score was almost one full standard
deviation above the mean. What about some of the
others?
9
Example 1 Calculating z-scores
79 81 80 77 73 83 74 93 78 80 75 67 73
77 83 86 90 79 85 83 89 84 82 77 72
Julia z(86-80)/6.07 z 0.99
above average z
6 7 7 2334 7 5777899 8 00123334 8
569 9 03
Kevin z(72-80)/6.07 z -1.32
below average -z
Katie z(80-80)/6.07 z 0
average z 0
10
Example 2 Comparing Scores

• Standardized values can be used to compare scores
  from two different distributions
• Statistics Test mean 80, std dev 6.07
• Chemistry Test mean 76, std dev 4
• Jenny got an 86 in Statistics and 82 in
  Chemistry.
• On which test did she perform better?

Although she had a lower score, she performed
relatively better in Chemistry.
11
Percentiles

• Another measure of relative standing is a
  percentile rank
• pth percentile Value with p of observations
  below it
• median 50th percentile mean50th ile if
  symmetric
• Q1 25th percentile
• Q3 75th percentile

What is Jennys Percentile?
6 7 7 2334 7 5777899 8 00123334 8
569 9 03
Jenny got an 86. 22 of the 25 scores are
86. Jenny is in the 22/25 88th ile.
12
Chebyshevs Inequality

• The of observations at or below a particular
  z-score depends on the shape of the distribution.
• An interesting (non-AP topic) observation
  regarding the of observations around the mean
  in ANY distribution is Chebyshevs Inequality.

Note Chebyshev only works for k gt 1
13
Summary and Homework

• Summary
• An individual observations relative standing can
  be described using a z-score or percentile rank
• We can describe the overall pattern of a
  distribution using a density curve
• The area under any density curve 1. This
  represents 100 of observations
• Areas on a density curve represent of
  observations over certain regions
• Homework
• Day 1 pg 118-9 probs 2-2, 3, 4,
  pg 122-123 probs 2-7, 8

14
Density Curve

• In Chapter 1, you learned how to plot a dataset
  to describe its shape, center, spread, etc
• Sometimes, the overall pattern of a large number
  of observations is so regular that we can
  describe it using a smooth curve

15
Density Curves

• Density Curves come in many different shapes
  symmetric, skewed, uniform, etc
• The area of a region of a density curve
  represents the of observations that fall in
  that region
• The median of a density curve cuts the area in
  half
• The mean of a density curve is its balance
  point

16
Describing a Density Curve

• To describe a density curve focus on
• Shape
• Skewed (right or left direction toward the
  tail)
• Symmetric (mound-shaped or uniform)
• Unusual Characteristics
• Bi-modal, outliers
• Center
• Mean (symmetric) or median (skewed)
• Spread
• Standard deviation, IQR, or range

17
Mean, Median, Mode

• In the following graphs which letter represents
  the mean, the median and the mode?
• Describe the distributions

18
Mean, Median, Mode

• (a) A mode, B median, C mean
• Distribution is slightly skewed right

• (b) A mean, median and mode (B and C
  nothing)
• Distribution is symmetric (mound shaped)

• (c) A mean, B median, C mode
• Distribution is very skewed left

19
Uniform PDF

• Sometimes we want to model a random variable that
  is equally likely between two limits
• When every number is equally likely in an
  interval, this is a uniform probability
  distribution
• Any specific number has a zero probability of
  occurring
• The mathematically correct way to phrase this is
  that any two intervals of equal length have the
  same probability
• Examples
• Choose a random time the number of seconds past
  the minute is random number in the interval from
  0 to 60
• Observe a tire rolling at a high rate of speed
  choose a random time the angle of the tire
  valve to the vertical is a random number in the
  interval from 0 to 360

20
Uniform Distribution

• All values have an equal likelihood of occurring
• Common examples 6-sided die or a coin

This is an example of random numbers between 0
and 1 This is a function on your calculator
Note that the area under the curve is still 1
21
Discrete Uniform PDF
P(x0) 0.25 P(x1) 0.25 P(x2) 0.25 P(x3)
0.25
Continuous Uniform PDF
P(x1) 0 P(x 1) 0.33 P(x 2) 0.66 P(x
3) 1.00
22
Example 1

• A random number generator on calculators randomly
  generates a number between 0 and 1. The random
  variable X, the number generated, follows a
  uniform distribution
• Draw a graph of this distribution
• What is the percentage (0ltXlt0.2)?
• What is the percentage (0.25ltXlt0.6)?
• What is the percentage gt 0.95?
• Use calculator to generate 200 random numbers

0.20
0.35
0.05
Math ? prb ? rand(200) STO L3 then 1varStat L3
23
Statistics and Parameters

• Parameters are of Populations
• Population mean is µ
• Population standard deviation is s
• Statistics are of Samples
• Sample mean is called x-bar or x
• Sample standard deviation is s

24
Summary and Homework

• Summary
• We can describe the overall pattern of a
  distribution using a density curve
• The area under any density curve 1. This
  represents 100 of observations
• Areas on a density curve represent of
  observations over certain regions
• Median divides area under curve in half
• Mean is the balance point of the curve
• Skewness draws the mean toward the tail
• Homework
• Day 2 pg 128-9 probs 2-9, 10, 12, 13,
  pg 131-133 probs 15, 18