Chapter 4 Displaying and Summarizing Quantitative Data Math2200

Example: Tsunamis and Earthquake - The most disastrous tsunami - Dec 26, 2004, in Sumatra - Earthquake: magnitude Killed 225,000 people

Question Was the earthquake that caused it truly unusually big? US National Geophysical Data Center Data on the magnitude of underlying earthquakes for 1240 historical tsunamis How do we learn the data YearCountryMagnitude -1300GREECE6 -479GREECE GREECE GREECE GREECE7 -227GREECE ALBANIA CYPRUS ISLAND NEW ZEALAND JAPAN JAPAN USA INDONESIA JAPAN JAPAN INDONESIA9

Histogram Display a quantitative variable by discretizing it into equal-width bins Counts for the bins give the distribution of the quantitative variable Make a bar chart based on these counts and align the bar according to the bin values, we get a histogram –Do not leave gaps between bars

With different number of bins

Summarize the histogram Magnitudes are typically around 7 Most are between 5.5 and 8.5 Minimum is around 3 Maximum is around 9 Why there is a sharp peak in the middle?

How to make a histogram using TI-83? Input the data –Suppose our data is 1.34, 2.21, 3.64, 6.2 –2nd + ( , 2.21, 3.64, nd + ) + STO + 2 nd + 1 –That gives L1 = {1.34, 2.21, 3.64, 6.2} Make the histogram –Press 2 nd +Y= (STAT PLOT) –You will see STAT PLOTS 1: Plot 1 … Off –Move the cursor to highlight 1:, then press ENTER –Select the following: ON Type: histogram (3 rd one) Xlist: L1 –Press ZOOM –Press 9 To adjust the graph window, press WINDOW and change the options

Stem-and-Leaf Plot John W. Tukey Useful for small data sets Similar to histogram, but the bars give numerical values more than counts 5 | 6 6 | | | | | | 8 Pulse-rates of 24 woman (8|8 means 88 beats/min)

Handwriting Handwriting may not give the same space for different digits. That violates the area principle When you make a stem-and-leaf plot, be sure to give each digit the same width.

Dotplot Replace digits in stem-and-leaf plot by dots

How to summarize the distribution of a quantitative variable? shape mode, symmetry, outlier center mean, median Spread sd, IQR

Shape Peak / Mode –Is there a peak? If so, how many peaks? –For quantitative variables, the mode is where the peak is at. –No peak: uniform –One peak: unimodal –Two peaks: bimodal –More than two peaks: multimodal

Shape Symmetry –Tail: thinner ends of a distribution –Skewed: If one tail stretches out farther than the other, we say the histogram is skewed to the side of the longer tail

Shape Outliers –Those that stand away from the body of the distribution –The judgment is vague sometimes

Center When a histogram is symmetric and unimodal, the center is obvious –The corresponding numerical value can be taken as the sample average, or say the sample mean –The sample mean is actually where the histogram balances

Center For skewed distribution –The sample mean is dragged to the side of the longer tail –Usually, much more than 50% values will be less or larger than the sample mean –Median is more appropriate Median is the value that splits the data in half

Finding the median Suppose that we have n numbers Order them first –If n is odd, the median is middle value. That is, the value in the (n+1)/2 position –If n is even, we take median as the average of the values in positionsn/2 and n/2+1

Mean versus median Extreme values / outliers: –Median only considers the order of the values, so it is resistant to extreme values –Mean is very sensitive Skewed distribution –Median is preferred than mean Unimodal and symmetric distribution –Mean is preferred because it uses more information from the data

Spread To quantify the variation Range Interquartile range (IQR) Standard deviation

Range Range = max – min Very sensitive to extreme values

Interquartile Range Quartiles –Q1 (lower quartile or the 25 th percentile): one quarter of the data lies below Q1 –Q2 (median or the 50 th percentile) –Q3 (upper quartile or the 75 th percentile): one quarter of the data lies above Q3 IQR = Q3-Q1 –Not sensitive to extreme values How to find Q1 and Q3? –Split the order values into two halves using the median –Q1 is the median of the first half –Q3 is the median of the second half

Standard deviation Sample variance = average of squared deviations Standard deviation (sd) –Sensitive to extreme values

σ X in TI-83

How to obtain these numbers using TI-83? Press STAT Move the cursor to CALC Press 1 The screen shows 1-Var Stats Put the list you want the statistics for. For example, L1. Press ENTER, then you will see –Sample mean, sample sum, sample sum squares, sample standard deviation (S x ),σ x (the same as except divided by n instead of n-1), sample size n, minimum, Q1, median, Q3, maximum

Summary Make a picture –Histogram, stem-and-leaf plot, dot plot Shape –How many modes? –Symmetric? –Outliers? If there are outliers, summarize once with the outliers and another time without the outliers Center and spread –Skewed distribution: median and IQR –Symmetric and unimodal distribution: mean and sd

What can go wrong? Do not use what we learned in chapter 4 for a categorical variable –Do not make histogram of a categorical variable –Do not look for shape and center and spread of a bar chart –Do not use mean, sd, IQR, etc. for a categorical variable Graph with bars are not always histograms or bar charts Choose a bin width appropriate to the data Check the summary numbers. Do they make sense? Do not worry about small differences when using different methods –No need to use too many digits for the summary numbers –Using one or two more digits than data is enough Do not round in the middle of a calculation Multiple modes, outliers (make a picture)