1. Chapter 2: Describing Location in a Distribution
Section 2.1
Measures of Relative Standing and Density Curves
The Practice of Statistics, 3rd edition - For AP*
STARNES, YATES, MOORE

2. Test Corrections

3. Descriptive Statistics
Visuals Graphs
Dot Plots
Histograms
Bar graphs
Box Plots
Stem and leaf
Numerical summaries
IQR and Median
Standard Deviation and Mean
5 number summaries
Normal
Skewed left or right
unimodal

4. Section 2.1 Describing Location in a Distribution
Learning Objectives
After this section, you should be able to…
MEASURE position using percentiles
MEASURE position using z-scores
TRANSFORM data
DEFINE and DESCRIBE density curves

5. Sample Data
Consider the following test scores for a small class: 79 81 80 77 73 83 74 93 78 75 67 86 90 85 89 84 82 72
Jenny’s score is noted in red. How did she perform on this test relative to her peers?
6 | 7
7 | 2334
7 |
8 |
8 | 569
9 | 03
6 | 7
7 | 2334
7 |
8 |
8 | 569
9 | 03
Her score is “above average”...
but how far above average is it?

6. Describing Location in a Distribution
Measuring Position: z-Scores
A z-score tells us how many standard deviations from the mean an observation falls, and in what direction.
Describing Location in a Distribution
Definition:
If x is an observation from a distribution that has known mean and standard deviation, the standardized value of x is:
A standardized value is often called a z-score.
Jenny earned a score of 86 on her test. The class mean is 80 and the standard deviation is What is her standardized score?

7. Describing Location in a Distribution
Using z-scores for Comparison
Describing Location in a Distribution
We can use z-scores to compare the position of individuals in different distributions.
Example
Jenny earned a score of 86 on her statistics test. The class mean was 80 and the standard deviation was She earned a score of 82 on her chemistry test. The chemistry scores had a fairly symmetric distribution with a mean 76 and standard deviation of 4. On which test did Jenny perform better relative to the rest of her class?

8. Density Curves, Empirical Rule & Normality, Z-score Intro
Homework #13 Pg 118 #1-4
Density Curves, Empirical Rule & Normality, Z-score Intro

9. Homework #13 Pg 118 #1-4

11. Describing Location in a Distribution
Density Curves
In Chapter 1, we developed a kit of graphical and numerical tools for describing distributions. Now, we’ll add one more step to the strategy.
Describing Location in a Distribution
Exploring Quantitative Data
Always plot your data: make a graph.
Look for the overall pattern (shape, center, and spread) and for striking departures such as outliers and gaps.
Calculate a numerical summary to briefly describe center and spread.
4. Sometimes the overall pattern of a large number of observations is so regular that we can describe it by a smooth curve.

12. Describing Location in a Distribution
Density Curve
Definition:
A density curve is a curve that
is always on or above the horizontal axis, and
has area exactly 1 underneath it.
A density curve describes the overall pattern of a distribution. The area under the curve and above any interval of values on the horizontal axis is the proportion of all observations that fall in that interval.
Describing Location in a Distribution
The overall pattern of this histogram of the scores of all 947 seventh-grade students in Gary, Indiana, on the vocabulary part of the Iowa Test of Basic Skills (ITBS) can be described by a smooth curve drawn through the tops of the bars.

13. 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 “equal area point”
The mean of a density curve is its “balance point”

14. 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)
Outliers and gaps
Center
Mean (symmetric) or median (skewed)
Spread
Standard deviation, IQR, or range

15. Mean and Median
In the following graphs which letter represents the mean and the median?
Describe the distributions

16. Mean, Median, Mode (a) B: median, C: mean
Distribution is slightly skewed right
(b) A: mean and median (B and C – nothing)
Distribution is symmetric (mound shaped)
(c) A: mean, B: median
Distribution is very skewed left

17. 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 (0<X<0.2)?
What is the percentage (0.25<X<0.6)?
What is the percentage > 0.95?
Use calculator to generate 200 random numbers 1
0.20
0.35
0.05
Math  prb  rand(200) STO L3 then 1varStat L3

18. Reminder Statistics Symbols
Parameters are of Populations
Population mean is μ (mu)
Population standard deviation is σ (omega)
Statistics are of Samples
Sample mean is called x-bar or x
Sample standard deviation is s

19. HW #14
Change HW sheet Pg 121 #5, Pg 128 #10,13,14

20. Density Curves, Empirical Rule & Normality, Z-score Intro
Homework #14 Pg 121 #5 Pg 128 #10,13,14
Density Curves, Empirical Rule & Normality, Z-score Intro

23. Density Curves, Empirical Rule & Normality, Z-score Intro
Homework #16 pg 131 #16,17,19,20 Review
Density Curves, Empirical Rule & Normality, Z-score Intro