Chapter 2 review: Graphical methods for frequency ...

1 Chapter 2 review: Graphical methods for frequency distributions Type of Data Method Categorical data Bar graph Numerical data Histogram Cumulative frequency distribution Chapter 2 review: Graphical methods for frequency distributions Type of Data Method Categorical data Bar graph Numerical data Histogram Cumulative frequency distribution Area (thousands of square miles)Frequency050001000015000010203040 Categorical Numerical Categorical Contingency table Grouped bar graph Mosaic plot Numerical Strip chart Box plot Multiple histograms Scatter plot Chapter 2 review: Graphical methods for associations between variables Categorical Numerical Categorical Contingency table Grouped bar graph Mosaic plot Numerical Strip chart Box plot Multiple histograms Scatter plot Chapter 2 review: Graphical methods for associations between variables Categorical Numerical Categorical Contingency table Grouped bar graph Mosaic plot Numerical Strip chart Box plot Multiple histograms Scatter plot Chapter 2 review: Graphical methods for associations between variables Describing data Two common descriptions of numerical data Location (or central tendency) Width (or spread) Measures of location Mean Median Mode Mean Y =Yii=1n nn is the size of the sample!

2 Mean Y1=56, Y2=72, Y3=18, Y4= 42 = (56+72+18+42) / 4 = 47 Y Median The median is the middle measurement in a set of ordered data. The data: 18 28 24 25 36 14 34 can be put in order: 14 18 24 25 28 34 36 Median is 25 The data: 18 28 24 25 36 14 34 17 can be put in order: 14 17 18 24 25 28 34 36 Median is FrequencyHeight (in cm) of Bio300 StudentsMode The mode is the most frequent measurement. Genotype Mean Median MM 63 Mm 59 mm 11 How do mean and median compare? The mean is the center of gravity; "the median is the middle measurement."The mean is more sensitive to extreme observations than the median"Mean and median for US household income, 2005 Median $46,326 Mean $63,344 Mode $5000-$9999 Why?

3 "Mean cmMedian 170 cmMode 165-170 cmUniversity student heights"Measures of width Range Standard deviation Variance Coefficient of variation Interquartile range Range 14 17 18 20 22 22 24 25 26 28 28 28 30 34 36 The range is the maximum minus the minimum: 36 -14 = 22 The range is a poor measure of distribution width Small samples tend to give lower estimates of the range than large samples So sample range is a biased estimator of the true range of the population."Variance in a population 2=Yi ()2i=1N NN is the number of individuals in the population." is the true mean of the population."Sample variance s2=Yi Y ()2i=1n n 1n is the sample size!Example: Sample variance s2=Yi Y ()2i=1n n 1 Family sizes of 5 BIOL 300 students: 2 3 3 4 4 Y =2+3+3+4+4()5=165= 3 3 4 4 16 Yi Y Yi (Yi Y )2 Sums:" s2= Sum of squares "(in units of siblings squared)"(in units of siblings)"Shortcut for calculating sample variance s2=nn 1# $ % & Yi2()i=1n n Y 2# $ ( ( ( % & ) ) ) Example: Sample variance (shortcut) Family sizes of 5 BIOL 300 students: 2 3 3 4 4 Y =2+3+3+4+4()5= 4 3 9 3 9 4 16 4 16 16 54 Yi Y Yi (Yi Y )2 Sums:" s2=54545 ()2# $ % & ' ( = s2=nn 1# $ % & ' ( Yi2()i=1n n Y 2# $ % % % % & ' ( ( ( ( Yi2 Standard deviation (SD) Positive square root of the variance is the true standard deviation"s is the sample standard deviation:!))))))

4 S=s2=Yi Y ()2i=1n n 1 s= s2= of variation (CV) CV=100%sY Interquartile Range Extreme values on box plots Values lying farther from the box edge than times the interquartile range Skew Skew is a measurement of asymmetry Skew (as in "skewer ) refers to the pointy tail of a distribution Nomenclature Population Parameters Sample Statistics Mean Variance s2 Standard Deviation s Y 2 Proportion = number in category n One common description of categorical data BrownBlueHazelGreenEye ColorFrequency050100150200 pEye Color Proportion Brown 220 / 592 = Blue 215 / 592 = Hazel 93 / 592 = Green 64 / 592 = 9 14 4 7 2 18 2 Calculate Mean, Median and Mode 9 14 4 7 2 18 2 Mean: 9 + 14 + 4 + 7 + 2 + 18 + 2 = 56 56 / 7 = 8 Calculate Mean, Median and Mode 9 14 4 7 2 18 2 Mean: 9 + 14 + 4 + 7 + 2 + 18 + 2 = 56 56 / 7 = 8 Median: 2 2 4 7 9 14 18 Calculate Mean, Median and Mode Calculate Mean, Median and Mode 9 14 4 7 2 18 2 Mean: 9 + 14 + 4 + 7 + 2 + 18 + 2 = 56 56 / 7 = 8 Median: 2 2 4 7 9 14 18 Mode: 2 2 4 7 9 14 18 Calculate Variance and Standard Deviation 9 14 4 7 2 18 2 = 8 s2=Yi Y ()2i=1n n 1 Variance:"Standard deviation:"Y2 -6 36 2 -6 36 4 -4 16 7 -1 1 9 1 1 14 6 36 18 10 100 56 226 Yi Y Yi (Yi Y )2 Sums:" = 8"!

5 S2 = 226 / 6 = ""s =" = "Calculate Variance and Standard Deviation are the units? 9 14 4 7 2 18 2 (cm) Mean: 8 cm Median: 7 cm Mode: 2 cm Variance: cm2 Standard Deviation: cm Readings We ve now covered Ch. 1-3 Next lecture we ll cover Ch. 4 Assignment Chapter 1: 14, 17, 19, 20 Chapter 2: 25, 32 Due Friday the 25th by 2 pm in your Ta s lock box Edition 1 users: Use problems on website! Wrong questions were briefly posted, so if you got them before Tuesday s class, go back and check that they are the correct ones!