1. Introduction to Biostatistics and Bioinformatics Exploring Data and Descriptive Statistics

2. Learning Objectives Python matplotlib library to visualize data: Scatter plot Histogram Kernel density estimate Box plots Descriptive statistics: Mean and median Standard deviation and inter quartile range Central limit theorem

3. An Example Data Set 0.022 -0.083 0.048 -0.010 -0.125 0.195 -0.071 -0.147 0.033 0.080 0.073 0.016 0.148 0.135 0.006 -0.089 0.165 -0.088 -0.137 0.094

4. Scatter Plot 0.022 -0.083 0.048 -0.010 -0.125 0.195 -0.071 -0.147 0.033 0.080 0.073 0.016 0.148 0.135 0.006 -0.089 0.165 -0.088 -0.137 0.094 Order or Measurement Measurement

5. Histogram Order or Measurement Measurement Bin size = 0.1Bin size = 0.05Bin size = 0.025 Number of Measurements

6. Cumulative Distributions Order or Measurement Measurement Cumulative Frequency

7. Kernel Density Estimate Order or Measurement Measurement Number of Measurements

8. Original Distribution Order or Measurement Measurement Number of Measurements Original Distribution Kernel Density Estimate Frequency Measurement Bin size = 0.05 Number of Measurements Histogram Measurement

9. More Data Order or Measurement Measurement Number of Measurements Original Distribution Kernel Density Estimate Frequency Measurement Bin size = 0.05 Number of Measurements Histogram Measurement

10. Exercise 1 Download ibb2015_7_exercise1.py (a)Draw 20 points from a normal distribution with mean=0 and standard deviation=0.1. import numpy as np y=0.1*np.random.normal(size=20) print y [-0.09946073 -0.19612617 0.03442682 0.02622746 -0.28418124 -0.04245968 0.05922837 0.01199874 0.13454915 -0.07482707 -0.11688758 0.01714036 0.03280043 0.01356022 0.09128649 -0.18923468 0.14536047 -0.07764629 -0.0349553 0.04300367]

11. Exercise 1 (b)Make scatter plot of the 20 points. import matplotlib.pyplot as plt x=range(1,points+1) fig, (ax1) = plt.subplots(1,figsize=(6,6)) ax1.scatter(x,y,color='red',lw=0,s=40) ax1.set_xlim([0,points+1]) ax1.set_ylim([-1,1]) fig.savefig('ibb2015_7_exercise1_scatter_points'+str(poi nts)+'.png',dpi=300,bbox_inches='tight') plt.close(fig)

12. Exercise 1 (c)Plot histograms. for bin in [20,40,80]: fig, (ax1) = plt.subplots(1,figsize=(6,6)) ax1.hist(y,bins=bin,histtype='step',color='black', range=[-1,1], lw=2, normed=True) ax1.set_xlim([-1,1]) fig.savefig('ibb2015_7_exercise1_bin'+str(bin)+'_ points'+str(points)+'.png',dpi=300,bbox_inches='t ight') plt.close(fig)

13. Exercise 1 (d)Plot cumulative distribution. y_cumulative=np.linspace(0,1,points) x_cumulative=np.sort(y) fig, (ax1) = plt.subplots(1,figsize=(6,6)) ax1.plot(x_cumulative,y_cumulative,color='black', lw=2) ax1.set_xlim([-1,1]) ax1.set_ylim([0,1]) fig.savefig('ibb2015_7_exercise1_cumulative_points'+ str(points)+'.png',dpi=300,bbox_inches='tight') plt.close(fig)

14. Exercise 1 (e)Plot kernel density estimate. import scipy.stats as stats kde_points=1000 kde_x = np.linspace(-1,1,kde_points) fig, (ax1) = plt.subplots(1,figsize=(6,6)) kde_y=stats.gaussian_kde(y) ax1.plot(kde_x,kde_y(kde_x),color='black', lw=2) ax1.set_xlim([-1,1]) fig.savefig('ibb2015_7_exercise1_kde_points'+str(points) +'.png',dpi=300,bbox_inches='tight') plt.close(fig)

15. Comparing Measurements

16. Comparing Measurements – Cumulative distributions

17. Systematic Shifts

18. Exercise 2 Download ibb2015_7_exercise2.py (a)Generate 5 data sets with 20 data points each from normal distributions with means = 0, 0, 0.1, 0.5 and 0.3 and standard deviation=0.1. y=[] for j in range(5): y.append(0.1*np.random.normal(size=20)) y[2]+=0.1 y[3]+=0.5 y[4]+=0.3 print y

19. Exercise 2 (b)Make scatter plots for the 5 data sets. sixcolors=['#D4C6DF','#8968AC','#3D6570','#91732B', '#963725','#4D0132'] fig, (ax1) = plt.subplots(1,figsize=(6,6)) for j in range(5): ax1.scatter(np.linspace(j+1-0.2,j+1+0.2,20), y[j],color=sixcolors[6-(j+1)], lw=0, alpha=1) ax1.set_xlim([0,6]) ax1.set_ylim([-1,1]) fig.savefig('ibb2015_7_exercise2_scatter_sample'+ str(20),dpi=300,bbox_inches='tight') plt.close(fig)

20. Correlation Between Two Variables

25. Data Visualization http://blogs.nature.com/methagora/2013/07/data -visualization-points-of-view.html

26. Process of Statistical Analysis Population Random Sample Sample Statistics Describe Make Inferences

27. Distributions ComplexNormalSkewedLong tails n=3 n=10 n=100

28. Mean Sample

29. Mean - Sample Size Normal Distribution 100 0.2 0.0 Mean 806040200 Sample Size -0.2

30. Mean – Sample Size ComplexNormalSkewedLong tails Sample Size 100 1 0.2 -0.2

31. Mode, Maximum and Minimum Sample Maximum Minimum Mode the most common value

32. Median, Quartiles and Percentiles Sample Quartiles for 25% of the sample for 50% of the sample (median) for 75% of the sample for m% of the sample Percentiles

33. Median and Mean – Sample Size ComplexNormalSkewedLong tails Sample Size 100 1 0.2 -0.2 Median - Gray

34. Variance Sample Mean

35. Variance – Sample Size ComplexNormalSkewedLong tails Sample Size 100 0.6 0 0.1 0

36. Inter Quartile Range (IQR) Sample Quartiles for 25% of the sample for 50% of the sample (median) for 75% of the sample Inter Quartile Range

37. Inter Quartile Range and Standard Deviation ComplexNormalSkewedLong tails Sample Size 100 1.0 0 0.4 0 IRQ/ 1.349 - Gray

38. Central Limit Theorem The sum of a large number of values drawn from many distributions converge normal if: The values are drawn independently; The values are from the one distribution; and The distribution has to have a finite mean and variance.

39. Uncertainty in Determining the Mean ComplexNormalSkewedLong tails n=3 n=10 Mean n=100 n=3 n=10 n=100 n=3 n=10 n=100 n=10 n=100 n=1000

40. Standard Error of the Mean Variance Sample Mean Standard Error of the Mean

41. Exercise 3 Download ibb2015_7_exercise3.py (a)Generate skewed data sets. sample_size=10 x_test=np.random.uniform(-1.0,1.0,size=30*sample_size) y_test=np.random.uniform(0.0,1.0,size=30*sample_size) y_test2=skew(x_test,-0.1,0.2,10) y_test2/=max(y_test2) x_test2=x_test[y_test<y_test2] x_sample=x_test2[:sample_size] 1.Generate a pair of random numbers within the range. 2.Assign them to x and y 3.Keep x if the point (x,y) is within the distribution. 4.Repeat 1-3 until the desired sample size is obtained. 5.The values x obtained in this was will be distributed according to the original distribution.

42. Exercise 3 (b)Calculate the mean of samples drawn from the skewed data set and the standard error of the mean, and plot the distribution of averages. for repeat in range(1000): … average.append(np.mean(x_sample)) sem=np.std(average) fig, (ax1) = plt.subplots(1,figsize=(6,6)) ax1.set_title('Sample size = '+str(sample_size)+', SEM = ' +str(sem)) ax1.hist(average,bins=100,histtype='step',color='red',range= [-0.5,0.5],normed=True,lw=2) ax1.set_xlim([-0.5,0.5])

43. Box Plot M. Krzywinski & N. Altman, Visualizing samples with box plots, Nature Methods 11 (2014) 119

44. n=5 Box Plots ComplexNormalSkewedLong tails n=10 n=100 n=5 n=10 n=100 n=5 n=10 n=100 n=5 n=10 n=100

45. Box Plots with All the Data Points ComplexNormalSkewedLong tails n=5 n=10 n=100 n=5 n=10 n=100 n=5 n=10 n=100 n=5 n=10 n=100

46. Box Plots, Scatter Plots and Bar Graphs Normal Distribution Error bars: standard deviation error bars: standard deviation error bars: standard error

47. Box Plots, Scatter Plots and Bar Graphs Skewed Distribution Error bars: standard deviation error bars: standard deviation error bars: standard error

48. Exercise 4 Download ibb2015_7_exercise4.py and plot box plots for a skewed data set. fig, (ax1) = plt.subplots(1,figsize=(6,6)) ax1.scatter(np.linspace(1-0.1, 1+0.1,sample_size), x_sample, facecolors='none', edgecolor=thiscolor, lw=1) bp=ax1.boxplot(x_samples, notch=False, sym='') plt.setp(bp['boxes'], color=thiscolor, lw=2) plt.setp(bp['whiskers'], color=thiscolor, lw=2) plt.setp(bp['medians'], color='black', lw=2) plt.setp(bp['caps'], color=thiscolor, lw=2) plt.setp(bp['fliers'], color=thiscolor, marker='o', lw=0) fig.savefig(…)

49. Descriptive Statistics - Summary Example distribution: Normal distribution Skewed distribution Distribution with long tails Complex distribution with several peaks Mean, median, quartiles, percentiles Variance, Standard deviation, Inter Quartile Range (IQR), error bars Box plots, bar graphs, and scatter plots

50. Descriptive Statistics – Recommended Reading http://blogs.nature.com/methagora/2013/08/giving_statistics_the_attention_it_deserves.html

51. Homework Plot the ratio of the standard error of the mean and the standard deviation as a function of sample size (use sample sizes of 3, 10, 30, 100, 300, 1000) for the skewed distribution in Exercise 3. Modify ibb2015_7_exercise3.py to generate this plot and email both the script and the plot.

52. Next Lecture: Sequence Alignment Concepts