# Lecture 2: Descriptive Statistics and Exploratory Data Analysis

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1 Lecture 2: Descriptive Statistics and Exploratory Data Analysis

2 Further Thoughts on Experimental Design 16 Individuals (8 each from two populations) with replicates Pop 1 Pop 2 Randomly sample 4 individuals from each pop Tissue culture and RNA extraction Labeling and array hybridization Slide scanning and data acquisition Repeat 2 times processing 16 samples in total Repeat entire process producing 2 technical replicates for all 16 samples

3 Other Business Course web-site: Homework due on Thursday not Tuesday Make sure you look at HW1 soon and see either Shameek or myself with questions

4 Today What is descriptive statistics and exploratory data analysis? Basic numerical summaries of data Basic graphical summaries of data How to use R for calculating descriptive statistics and making graphs

5 Central Dogma of Statistics Population Probability Descriptive Statistics Sample Inferential Statistics

6 EDA Before making inferences from data it is essential to examine all your variables. Why? To listen to the data: - to catch mistakes - to see patterns in the data - to find violations of statistical assumptions - to generate hypotheses and because if you don t, you will have trouble later

7 Types of Data Categorical Quantitative binary nominal ordinal discrete continuous 2 categories more categories order matters numerical uninterrupted

8 Dimensionality of Data Sets Univariate: Measurement made on one variable per subject Bivariate: Measurement made on two variables per subject Multivariate: Measurement made on many variables per subject

9 Numerical Summaries of Data Central Tendency measures. They are computed to give a center around which the measurements in the data are distributed. Variation or Variability measures. They describe data spread or how far away the measurements are from the center. Relative Standing measures. They describe the relative position of specific measurements in the data.

10 Location: Mean 1. The Mean x To calculate the average of a set of observations, add their value and divide by the number of observations: x = x 1 + x 2 + x x n n = 1 n n " x i i=1!

11 Other Types of Means Weighted means: x = n " i=1 n " i=1 w i x i w i Trimmed: x = "! Geometric: # x = % \$ n " i=1 x i & ( ' 1 n! Harmonic: x = n n 1 " i=1 x i!!

12 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 middle two values and average them Example Some data: Age of participants: Median = (22+23)/2 = 22.5

13 Which Location Measure Is Best? Mean is best for symmetric distributions without outliers Median is useful for skewed distributions or data with outliers Mean = 3 Mean = 4 Median = 3 Median = 3

14 Scale: Variance Average of squared deviations of values from the mean " ˆ 2 = n \$ i (x i # x ) 2 n #1!

15 Why Squared Deviations? Adding deviations will yield a sum of? Absolute values do not have nice mathematical properties Squares eliminate the negatives Result: Increasing contribution to the variance as you go farther from the mean.

16 Scale: Standard Deviation Variance is somewhat arbitrary What does it mean to have a variance of 10.8? Or 2.2? Or ? Or ? Nothing. But if you could standardize that value, you could talk about any variance (i.e. deviation) in equivalent terms Standard deviations are simply the square root of the variance

17 Scale: Standard Deviation ˆ " = n \$ i (x i # x ) 2 n #1 1. Score (in the units that are meaningful) 2. Mean! 3. Each score s s deviation from the mean 4. Square that deviation 5. Sum all the squared deviations (Sum of Squares) 6. Divide by n-1 7. Square root now the value is in the units we started with!!!

18 Interesting Theoretical Result Regardless of how the data are distributed, a certain percentage of values must fall within k standard deviations from the mean: Note use of µ (mu) to represent mean. At least Note use of σ (sigma) to represent standard deviation. within (1-1/1 2 ) = 0%... k=1 (μ ± 1σ) (1-1/2 2 ) = 75%... k=2 (μ ± 2σ) (1-1/3 2 ) = 89%...k=3 (μ ± 3σ)

19 Often We Can Do Better For many lists of observations especially if their histogram is bell-shaped 1. Roughly 68% of the observations in the list lie within 1 standard deviation of the average 2. 95% of the observations lie within 2 standard deviations of the average Ave-2s.d. Ave-s.d. Average Ave+s.d. Ave+2s.d. 68% 95%

20 Scale: Quartiles and IQR IQR 25% 25% 25% 25% Q 1 Q 2 Q 3 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, 50% are larger) Only 25% of the observations are greater than the third quartile

21 Percentiles (aka Quantiles) In general the n th percentile is a value such that n% of the observations fall at or below or it n% Q 1 = 25 th percentile Median = 50 th percentile Q 2 = 75 th percentile

22 Graphical Summaries of Data A (Good) Picture Is Worth A 1,000 Words

23 Univariate Data: Histograms and Bar Plots What s the difference 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 pictorial representation Histogram Used to visualize distribution (shape, center, range, variation) of continuous variables Bin size important

24 Effect of Bin Size on Histogram Simulated 1000 N(0,1) and 500 N(1,1) Frequency Frequency Frequency

25 More on Histograms What s the difference between a frequency histogram and a density histogram?

26 More on Histograms What s the difference between a frequency histogram and a density histogram? Frequency Histogram Density Histogram

27 100.0 Box Plots maximum 66.7 Q 3 Years 33.3 IQR Q 1 median minimum 0.0 AGE Variables

28 Bivariate Data Variable 1 Variable 2 Display Categorical Categorical Crosstabs Stacked Box Plot Categorical Continuous Boxplot Continuous Continuous Scatterplot Stacked Box Plot

29 Clustering Multivariate Data Organize units into clusters Descriptive, not inferential Many approaches Clusters always produced Data Reduction Approaches (PCA) Reduce n-dimensional dataset into much smaller number Finds a new (smaller) set of variables that retains most of the information in the total sample Effective way to visualize multivariate data

30 How to Make a Bad Graph The aim of good data graphics: Display data accurately and clearly Some rules for displaying data badly: Display as little information as possible Obscure what you do show (with chart junk) Use pseudo-3d and color gratuitously Make a pie chart (preferably in color and 3d) Use a poorly chosen scale From Karl Broman:

31 Example 1

32 Example 2

33 Example 3

34 Example 4

35 Example 5

36 R Tutorial Calculating descriptive statistics in R Useful R commands for working with multivariate data (apply and its derivatives) Creating graphs for different types of data (histograms, boxplots, scatterplots) Basic clustering and PCA analysis

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