Descriptive Univariate Statistics and Bivariate Correlation

Transcription

1 ESC 100 Exploring Engineering Descriptive Univariate Statistics and Bivariate Correlation Instructor: Sudhir Khetan, Ph.D. Wednesday/Friday, October 17/19, 2012

2 The Central Dogma of Statistics used to summarize data; (this is the focus for today) used to make inferences about the population

3 Dimensionality of data sets Univariate: measurements made on one variable per subject. Bivariate: measurement made on two variables per subject. Multivariate: measurement made on many variables per subject.

4 What types of descriptive statistics are there? Central tendency measures: computed to give a center around which the measurements in the data are distributed (also called measures of location. Variation or variability measures: describe data spread or how far the measurements are away from the center. Relative standing measures: describe the relative position of specific measurements in the data.

5 Location: mean The mean: To calculate the average x of a set of observations, add their value and divide by the numberof observations:

6 Location: median Median the exact middle value Calculation If there are an odd number of observations, find the middle value If there are an even number of observations, find the mean of the middle two values Example: Median = ave(22,23) = 22.5 Age of students:

7 Which location measure is better? Mean is best for symmetric distributions with outliers Median is useful for skewed distributions or data with outliers

8 Location: mode The mode is the observation that takes place most frequently in a data set Unlike the mean or median, the mode is not necessarily unique the same maximum frequency may occur at different values.

9 Scale: variance Average of squared deviations of values from the mean Square the deviations to get rid of the negatives The result is that the contribution to the variance increases as you go farther from the mean in either direction

10 Scale: standard deviation Procedure to obtain standard deviation: Score/measure observations (in the units that are meaningful, let s say m/s) Find the mean of the observations (m/s) Find each score s deviation from the mean (m/s) Square all those deviations (m/s) 2 Divide by n 1 (m/s) 2 (note that this is the variance) square root (m/s) now we have the starting units!

11 An interesting theoretical result Regardless of how the data was distributed, a certain percentage of values will always fall within k standard deviations from the mean:

12 Often we can do better For many lists of observations, especially if their histogram is bell shaped roughly 68% of values in the list lie within 1 standard deviation of the mean roughly 95% two standard deviations of the mean

13 Scale: quartiles and IQR The first quartile, Q 1, is the value for which 25% of the observations are smaller and 75% are larger Q 2 is the same as the median (50% are smaller, and 50% are larger) Only 25% of the observations are greater than the third quartile

14 Percentiles (aka quantiles) Generally, the n th percentile is a value such that n% of the observations fall at or below it: Q 1 = 25 th percentile Median = 50 th percentile Q 2 = 75 th percentile

15 Graphical summaries of data A (good) picture is worth 1,000 words!

16 Univariate data: histograms and bar plots What s the differences between a histogram and bar plot? Bar plot Used for categorical variables to show frequency or proportion in each category. Translate the data from frequency tables into a pictoral representation... Histogram Used to visualize distribution (shape, center, range, variation) of continuous variables bin size is important

17 Effect of bin size on histogram Simulated 1000 N(0,1) 1000 random numbers from the distribution with mean 0 and st. dev. 1

18 More on histograms What s the difference between a frequency histogram and a density histogram?

19 More on histograms What s the difference between a frequency histogram and a density histogram?

20 Stem and leaf plots

21 Box and whisker plots An outlier is a score either 1.5 IQR above the upper quartile or below the lower quartile

22 Example problem Two different classes take a quiz and gets the following scores. Class 1: 2, 4, 6, 8, 10, 12, 14 Class 2: 2, 2, 3, 8, 8, 10, 23 What the mean and median of each set? The same! Will making a box and whisker plot of each set of data give us a better picture of their distributions? (let s do the first one together)

23 Box plot procedure Steps to make our box plot: Find the median, Q1, Q3, and IQR Draw 3 horizontal lines, at Q1, median, and Q3 Draw the corresponding vertical lines to make the boxes Compute the lower inner fence (Q1 1.5*IQR) and the upper inner fence (Q *IQR) Draw a whisker downward from Q1 to lower inner fence or minimum, whichever comes first Draw a whisker upward from Q3 to upper inner fence or maximum, whichever comes first Compute the lower outer fence (Q1 3*IQR) and the upper outer fence (Q3 + 3*IQR) Mild outliers fall between the inner and outer fences, mark with O Extreme outliers fall outside outer fences, mark with *

24 Scatter plots and correlation A scatter plot (or scatter diagram) is used to show the relationship between two variables Correlation analysis is used to measure strength of the association (linear relationship) between two variables only concerned with strength of the relationship No causal effect is implied or explored

25 Scatter plot examples Strong relationships Weak relationships y y x x y y x x

26 Scatter plot examples (cont.) No relationship y y x x

27 Example of a correlation in life Name a few others?

28 Correlation coefficient The population correlation coefficient ρ (rho) measures the strength of the association between the variables The sample correlation coefficient r is an estimate of ρ and is used to measure the strength of the linear relationship in the sample observations

29 Features of ρ and r Unit free Range between 1 and -1 The closer to -1, the stronger the negative linear relationship The closer to 1, the stronger the positive linear relationship The closer to 0, the weaker the linear relationship

30 Examples of approximate r values y y y x r = -1 r = -.6 r = 0 y y x x x r = +.3 r = +1 x

31 Calculating the correlation coefficient Sample correlation coefficient: r = [ (x (x x)(y y) x) 2 ][ (y y) 2 ] or the algebraic equivalent: where: r = [n( n xy x y x ) ( x) ][n( y ) ( r = Sample correlation coefficient n = Sample size x = Value of the independent variable y = Value of the dependent variable y) 2 ]

32 Calculation example Tree Height Trunk Diameter y x xy y 2 x Σ= Σ= Σ= Σ= Σ=

33 Calculation example (cont.) Tree Height, y 70 r = n xy x y [n( x ) ( x) ][n( y ) ( y) 2 ] = 8(3142) (73)(321) [8(713) (73) 2 ][8(14111) (321) 2 ] = Trunk Diameter, x r = relatively strong positive linear association between x and y

34 References Lecture 2 Descriptive Statistics and Exploratory Data Analysis University of Washington School of Medicine. Introduction to Linear Regression and Correlation Analysis Fordham University s/math_300/final/p14/default.html