Lecture 5: Measures of Center and Variability for Distributions (Population); Quartiles, Boxplots for Data (Sample) Chapter 2

Transcription

1 Lecture 5: Measures of Center and Variability for Distributions (Population); Quartiles, Boxplots for Data (Sample) Chapter 2

2 2.1 Measures of Center (Distributions) Mean for continuous distributions Let f(x) be the density function for a continuous random variable X, then the mean of X is: µ X x f ( x) = dx Mean for discrete distributions Let p(x) be the mass function for a continuous random variable X, then the mean of X is: µ X = x p(x)

3 Examples Means Continuous distributions Normal (µ,σ) µ Exponential (λ) 1/ λ Uniform (a, b) (a+b)/2 Discrete distributions Binomial (n, π) nπ Poisson (λ) λ

4 Example 1 Find the mean value of a variable x with density function: f(x) = 1.5(1-x^2), 0<x<1 = 0, o.w. What s the median of X distribution?

5 Example - Medians Median for continuous distributions Let f(x) be the density function for a continuous random variable X, then the median of X is whatever value which satisfies: ~ µ f ( x) dx = 0. 5 What is the median of a Normal Distribution? If a continuous distribution is perfectly symmetric, mean = median. No median for discrete distributions.

6 2.2 Measures of Variability (Distributions) Variance for continuous distributions Let f(x) be the density function for a continuous random variable X, then the variance of X is: σ ( ) 2 2 X x µ X f ( x) = dx Variance for discrete distributions Let p(x) be the mass function for a continuous random variable X, then the variance of X is: σ X ( x ) 2 σ X = µ p( x) - Standard deviation of X, is square root of the 2 variance σ X X 2

7 Examples Variances Continuous distributions Normal (µ,σ) σ 2 Exponential (λ) 1/ λ 2 Uniform (a, b) (b a) 2 /12 Discrete distributions Binomial (n, π) nπ(1 π) Poisson (λ) λ Self-reading: empirical rule ( rule) in the middle of Pg 76, unbiasedness on top of Pg 77

8 Example 2 What s the standard deviation of X from Example 1?

9 2.3 Other measures Quartiles The median is the midpoint of the data (Sample) Quartiles break the data into quarters 1 st Quartile (Q1) = lower quartile = 25 th percentile 2 nd Quartile = median = 50 th percentile 3 rd Quartile (Q3) = upper quartile = 75 th percentile How to find the quartiles? They are just medians of the two halves of the data Interquartile Range (or IQR) = Q3 Q1 Self reading: Percentiles, Pg 85

10 Example Scores for 10 students are: Find the median and quartiles: 1. Median= Q2 = M = (82+83)/2 = Q1 = Median of the lower half, i.e , = Q3 = Median of the upper half, i.e , = 85 Therefore, IQR = Q3 Q1 = = 5 Additionally, find Min and Max Min = 78, and Max = 87 We get a five-number summary! Min Q1 Median Q3 Max

11 Boxplots; Modified Version Visual representation of the five-number summary Central box: Q1 to Q3 Line inside box: Median Extended straight lines: from each end of the box to lowest and highest observation. Modified Boxplots: only extend the lines to the smallest and largest observations that are not outliers. Each mild outlier* is represented by a closed circle and each extreme outlier** by an open circle. *Any observation farther than 1.5 IQR from the closest quartile is an outlier. **An outlier is extreme if more than 3 IQR from the nearest quartile, and is mild otherwise.

12 Example Five-number summary is: Min: 78 Q1: 80 Median: 82.5 Q3: 85 Max: 87 Draw a boxplot:

13 More on Boxplots Much more compact than histograms Quick and Dirty visual picture Gives rough idea on how data is distributed Shows center/typical value (the median); Position of median line indicates symmetric/not symmetric, positively/negatively skewed. IQR gives the middle 50% Min to Max gives the entire range Side-by-side boxplots very useful for comparisons See from slide 10

14 Describe a Boxplot Symmetric? if not, positively or negatively skewed (based on median line) Outliers? Based on 1.5IQR rule (and 3IQR rule for extreme outliers) Overall range : = Max - Min; IQR : = Central box s range; Similar procedure for side-by-side comparison

15 Examples--MPG

16 After Class Review sections 2.1 and 2.2, especially Pg 63 68, Review section 2.3, self-reading Pg95 Read section 2.4 and 5.1 Hw#2, due by 5pm, next Monday Start Lab#2 now, you don t want to wait till the last minute of next Thursday pm