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

Test Corrections

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

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

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 | 5777899 8 | 00123334 8 | 569 9 | 03 6 | 7 7 | 2334 7 | 5777899 8 | 00123334 8 | 569 9 | 03 Her score is “above average”... but how far above average is it?

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 6.07. What is her standardized score?

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 6.07. 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?

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

Homework #13 Pg 118 #1-4

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.

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.

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”

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

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

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

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

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

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

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

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