Chapter 4: Displaying & Summarizing Quantitative Data

Chapter 4: Displaying & Summarizing Quantitative DataAP StatisticsSummarizing the data will help us when we look at large sets of quantitative data.

Without summaries of the data, its hard to grasp what the data tell us.

The best thing to do is to make a picture

We cant use bar charts or pie charts for quantitative data, since those displays are for categorical variables.

Dealing With a Lot of NumbersLists classes (or categories) of values, along with frequencies (or counts) of the number of values that fall into each class

Lower class limits: the smallest numbers that belong to different classesUpper class limits: the largest numbers that belong to different classesClass midpoints: midpoints of the classes found by adding the lower and upper limits of each class and dividing by 2Class width: the difference between two consecutive lower class limits

Frequency Distributions for Quantitative Data1) Figure out class width:Class Width = Max # - Min # # of classes (btw 5-20)

Make it the next largest integer!

2) Set up Classes:Start the first class with Min #Add the class width to the Min # to get the next lower limit.Continue to do this until you have the number of classes that are required.Go back and create you upper limits (1 less than the next lower limit)Last upper limit Either add your class width to the previous upper limit or what would the upper limit be if there was another class.Creating Frequency Distributions for Quantitative Data3) Make Tallies:What class does the data piece fall into put a tally at that class.4) Count tallies and put the number in frequency column.

5) Find the midpoint of each class. Add the upper and lower class limits together and divide by 2.

6) Find the relative frequency for each class.The frequency in that class Total number of frequency

Creating Frequency Distributions for Quantitative Data (cont.)7) Find the Cumulative FrequencyThe sum of the frequency for that class and all the classes above.

Creating Frequency Distributions for Quantitative Data (cont.)Example: Create a frequency distribution on the data given about time (in minutes) spent reading the newspaper in a day.Data:739139258222218230765293113916158151235

Frequency Distribution ExampleClassesTalliesFrequencyMidpointRelative FrequencyCumulative FrequencyCumulative Relative FrequencySimilar to a bar chart for categorical data, histograms graph the frequency of classes of quantitative data.

Histogram bars touch, unlike bar charts. This separates the two.

Notice how histograms do not have spaces between the bars as bar charts do with categorical data. Any spaces in a histogram are actual gaps in the data where there are no values.

Histograms: Displaying the Distribution of Earthquake Magnitudes

A relative frequency histogram displays the percentage of cases in each bin instead of the count. In this way, relative frequency histograms are faithful to the area principle.

Histograms: Displaying the Distributionof Earthquake Magnitudes (cont.)

Enter your data into lists: STAT editEnter data into L1

Go to 2nd Y= ENTERMake sure the Statplot is ONHighlight the histogramDefine the list as L1

Zoom 9 will give you a nice window for a histogramCreate a Histogram Using the Graphing CalculatorWed like to compare the distributions of two historically great baseball hitters: Babe Ruth and Mark McGwire. We have the following information on the numbers of home runs hit each year by Baby Ruth from 1920 1934 and for McGwire from 1986 2001.

Create a histogram for each players number of homeruns. What do you see?

Create a Histogram Using the Graphing Calculator Example

Stem-and-leaf displays show the distribution of a quantitative variable, like histograms do, while preserving the individual values.

Stem-and-leaf displays contain all the information found in a histogram and, when carefully drawn, satisfy the area principle and show the distribution.

Stem-and-Leaf DisplaysDAY 2Compare the histogram and stem-and-leaf display for the pulse rates of 24 women at a health clinic. Which graphical display do you prefer? Stem-and-Leaf Example

First, cut each data value into leading digits (stems) and trailing digits (leaves). Use the stems to label the bins.Use only one digit for each leafeither round or truncate the data values to one decimal place after the stem.If your stem-and-leaf looks too crowded, separate leaves into two categories: 0-4 and 5-9.

Constructing a Stem-and-Leaf Display

Listed is the weight of 30 cattle (in pounds) on a local farm. Create a stem-and-leaf plot to display these data:

1367 1123 1997187612241455 10261855139612451233 1977132111901064143213991543164310231520 14881722167513971496 1874194318651674Example: Stem-and-Leaf DisplaysA dotplot is a simple display. It just places a dot along an axis for each case in the data.

The dotplot to the right shows Kentucky Derby winning times, plotting each race as its own dot.

You might see a dotplot displayed horizontally or vertically.

Dotplots

Remember the Make a picture rule? Now that we have options for data displays, you need to Think carefully about which type of display to make.Before making a stem-and-leaf display, a histogram, or a dotplot, check theQuantitative Data Condition: The data are values of a quantitative variable whose units are known.When describing a distribution, make sure to always tell about three things: shape, center, and spread

Quantitative Data ConditionDoes the histogram have a single, central peak or several separated peaks?

Is the histogram symmetric?

Do any unusual features stick out?

What is the Shape of the Distribution?Does the histogram have a single, central peak or several separated peaks?

Humps in a histogram are called modes.

A histogram with one main peak is dubbed unimodal; histograms with two peaks are bimodal; histograms with three or more peaks are called multimodal.

Histogram PeaksA bimodal histogram has two apparent peaks:Histogram Peaks (cont.)

A histogram that doesnt appear to have any mode and in which all the bars are approximately the same height is called uniform:

Histogram Peaks (cont.)

Is the histogram symmetric?If you can fold the histogram along a vertical line through the middle and have the edges match pretty closely, the histogram is symmetric.

Symmetry

The (usually) thinner ends of a distribution are called the tails. If one tail stretches out farther than the other, the histogram is said to be skewed to the side of the longer tail.In the figure below, the histogram on the left is said to be skewed left, while the histogram on the right is said to be skewed right.

Symmetry (cont.)

Do any unusual features stick out?Sometimes its the unusual features that tell us something interesting or exciting about the data.

You should always mention any stragglers, or outliers, that stand off away from the body of the distribution.

Are there any gaps in the distribution? If so, we might have data from more than one group.

Anything Unusual?The following histogram has outliersthere are three cities in the leftmost bar:Anything Unusual? (cont.)

If you had to pick a single number to describe all the data what would you pick?

Its easy to find the center when a histogram is unimodal and symmetricits right in the middle.

On the other hand, its not so easy to find the center of a skewed histogram or a histogram with more than one mode.

Where is the Center of the Distribution?The median is the value with exactly half the data values below it and half above it.

It is the middle data value (once the data values have been ordered) that divides the histogram into two equal areas

It has the same unitsas the data

Center of a Distribution Median

Variation matters, and Statistics is about variation.

Are the values of the distribution tightly clustered around the center or more spread out?

Always report a measure of spread along with a measure of center when describing a distribution numerically.mHow Spread Out is the Distribution?The range of the data is the difference between the maximum and minimum values:

Range = max min

A disadvantage of the range is that a single extreme value/outlier can make it very large and, thus, not representative of the data overall.

Spread: RangeThe interquartile range (IQR) lets us ignore extreme data values and concentrate on the middle of the data.

To find the IQR, we first need to know what quartiles are

Spread: The Interquartile RangeQuartiles divide the data into four equal sections. One quarter of the data lies below the lower quartile, Q1One quarter of the data lies above the upper quartile, Q3.The quartiles border the middle half of the data.

The difference between the quartiles is the interquartile range (IQR), so

IQR = upper quartile lower quartileSpread: The Interquartile Range (cont.)Find the range and IQR of these data:

1236441922271000342531Spread: The Interquartile Range (cont.)DAY 3The lower and upper quartiles are the 25th and 75th percentiles of the data, soThe IQR contains the middle 50% of the values of the distribution, as shown in figure:Spread: The Interquartile Range (cont.)

The 5-number summary of a distribution reports its median, quartiles, and extremes (maximum and minimum)The 5-number summary for the recent tsunami earthquake Magnitudes looks like this:

5-Number Summary

Enter your data into L1 (Stat Edit)

To clear data, be sure to highlight the title (L1) and press CLEAR Enter (DO NOT PRESS DELETE)

Once your data is entered, press STAT CALC 1-Variable Stat

5-Number Summary on Calculator:Below is the weight of pennies (note that they changed from copper to zinc in the early 1980s):

2.57 2.56 3.14 3.03 3.13 2.47 2.43 3.11 3.06 2.482.51 2.50 3.07 3.08 3.01 2.45 2.50 3.13 2.51 3.123.10 3.08 2.46 2.44 2.47 2.54 3.09 3.13 2.56 2.49

Create a graphical display of these data and create a 5-number summary. What can you TELL about these data?

Example: Penny WeightsWhen we have symmetric data, there is an alternative other than the median.If we want to calculate a number, we can average the data.We use the Greek letter sigma to mean sum and write:

Summarizing Symmetric Distributions The MeanThe formula says that to find the mean, we add up all the values of the variable and divide by the number of data values, n.

The mean feels like the center because it is the point where the histogram balances:

Summarizing Symmetric Distributions The Mean (cont.)

Because the median considers only the order of values, it is resistant to values that are extraordinarily large or small; it simply notes that they are one of the big ones or small ones and ignores their distance from center.

To choose between the mean and median, start by looking at the data. If the histogram is symmetric and there are no outliers, use the mean.

However, if the histogram is skewed or with outliers, you are better off with the median.

Mean or Median?A more powerful measure of spread than the IQR is the standard deviation, which takes into account how far each data value is from the mean.A deviation is the distance that a data value is from the mean. Since adding all deviations together would total zero (because the positives and negatives would cancel each other out), we square each deviation and find an average of sorts for the deviations.What About Spread? The Standard DeviationDAY 4The variance, notated by s2, is found by summing the squared deviations and (almost) averaging them:

Why are we ALMOST averaging them?

The variance will play a role later in our study, but it is problematic as a measure of spreadit is measured in squared units!What About Spread? The Standard Deviation (cont.)

The standard deviation, s, is just the square root of the variance and is measured in the same units as the original data. What About Spread? The Standard Deviation (cont.)

A class has been divided into groups of five students each. The groups have completed an independent study project, and at the end they take an individual 20-point quiz. Here are the scores, by group:

Find the mean, standard deviation, & range for each.

Example: Standard Deviation

Since Statistics is about variation, spread is an important fundamental concept of Statistics.

Measures of spread help us talk about what we dont know.

When the data values are tightly clustered around the center of the distribution, the IQR and standard deviation will be small.

When the data values are scattered far from the center, the IQR and standard deviation will be large.

Thinking About VariationWhen telling about quantitative variables, start by making a histogram or stem-and-leaf display and discuss the shape of the distribution.

Next, always report the shape of its distribution, along with a center and a spread.

If the shape is skewed, report the median and IQR.

If the shape is symmetric, report the mean and standard deviation and possibly the median and IQR as well.Recap: Shape, Center, SpreadIf there are multiple modes, try to understand why. If you identify a reason for the separate modes, it may be good to split the data into two groups.

If there are any clear outliers and you are reporting the mean and standard deviation, report them with the outliers present and with the outliers removed. The differences may be quite revealing.

Note: The median and IQR are not likely to be affected by the outliers.Recap: Unusual FeaturesDont make a histogram of a categorical variablebar charts or pie charts should be used for categorical data.

Dont look for shape, center, and spread of a bar chart. Order doesnt matter, so these wouldnt makesense.

What Can Go Wrong?

Choose a bin width appropriate to the data.Changing the bin width changes the appearance of the histogram:

What Can Go Wrong? (cont.)

Dont forget to do a reality check dont let the calculator do the thinking for you.Dont forget to sort the values before finding the median or percentiles.Dont worry about small differences when using different methods.Dont compute numerical summaries of a categorical variable.Dont report too many decimal places.Dont round in the middle of a calculation.Watch out for multiple modesBeware of outliersMake a picture make a picture . . . make a picture !!!What Can Go Wrong? (cont.)Day 1: # 9A, 39Day 2: # 5, 7, 8, 9B, 10, 13-14 (by hand/show work)Day 3: # 14 (calculator), 31A-C, 32, 37 ADay 4: # 11, 16 (by hand), 17, 30 (calculator)Day 5: # 19, 22, 24, 25Day 6: # 29, 36, 44, 49

Reading: Chapter 5

Assignments: pp. 72 79