Topic 2: Describing Data Graphically

Learning Goals

• Create and interpret graphs (dot plots, histograms, pie charts, bar charts) as a means of summarizing and communicating data meaningfully.

• Identify the shape of a distribution (right-skewed, left-skewed, symmetric, or uniform).

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1. Distribution of Numerical Data

• In data analysis, one goal is to describe patterns (known as the distribution) of the variable in the data set and create a useful summary about the set.
• To describe patterns in data, we use descriptions of shape, center, and spread. We also describe exceptions to the pattern. We call these exceptions outliers.

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2. Dot Plots

A dot plot includes all values from the data set, with one dot for each occurrence of an observed value from the set.

How to Construct

1. Draw a horizontal line and mark it with an appropriate measurement scale.
2. Locate each value in the data set along the measurement scale, and represent it by a dot. Stack the dots vertically if the value appear multiple times.

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3. Example: Petal Lengths of Iris Flower

The data set contains 15 petal lengths of iris flower. Create a dot plot to describe the distribution of petal lengths.

Solution: For each number in the data set, we draw a dot. We stack dots of the same value from bottom to up.

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Practice: Heights of Cherry Trees

The data set contains the heights of 20 Black Cherry Trees. Create a dot plot to describe the distribution of the heights.

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4. Histograms (1 of 2)

• Histograms are frequently used to describe large sets of numerical data, in particular, data of a continuous variable.

• A histogram consists of a horizontal axis, a vertical axis and adjoining vertical bars. The horizontal axis is labeled to break data values into bins (classes). The vertical axis is then scaled accordingly so that the area of a bar is the frequency or the relative frequency of values in the bin.

• To construct a histogram, we need to decide boundary of bars which forms intervals, called bins or classes and find the frequency of values in each bin.

• A frequency distribution is a table which contains bins, frequencies and relative frequencies which are proportions (percentage) defined by the formula \(\text{Relative frequency} =\dfrac{\text{Class frequency}}{\text{Sample size}}\).

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5. Histograms (2 of 2)

• The lower boundary of a bin is called a lower bin limit. The upper boundary of a bin is called an upper bin limit.

• The bin width is the distance between the lower (or upper) bin limits of two consecutive bins.

• The difference between the maximum and the minimum data entries is called the range.

• The range is roughly, indeed slightly smaller than, \(\text{number of bins}\times \text{bin width}\).

• The midpoint of a bin is the half of the sum of the lower and upper limits of the bin.

• Bin limits should be chosen so that each value of the data is in exactly one bin.

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6. Conventions on Constructing Bins

• For a continuous variable: take an appropriate bin width, take the first lower bin as the minimum, and then take left closed and right opened intervals as bins.
• For a discrete variable: take an appropriate bin width and take the first lower bin limit to be a value slightly smaller than the minimum minus of the data set.
• Excel convention (roughly): take left opened and right closed intervals.

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7. Example: Histogram of mpg (1 of 3)

The following data set show the mpg (mile per gallon) of \(30\) cars. Construct a frequency table and frequency histogram for the data set using \(7\) bins. What can be concluded from the histogram?

Solution:

• Find the range using \(\text{Range}=\text{maximum}-\text{minimum}\). Here, the minimum is \(10.4\), the maximum is \(33.9\), and the range is \(33.9-10.4=23.5\).
• For \(k\) bins, the bin width can be taken between \(\frac{\mathrm{range}}{k}\) and \(\frac{\mathrm{range}}{k-1}\). Here, we take the bin width as \(3.4\) which is between \(\frac{23.5}{7}\approx 3.357\) and \(\frac{23.5}{7-1}\approx 3.917\).
• Take the first lower bin limit as the minimum \(10.4\). Add the bin width \(3.4\) recursively to get all lower bin limits: 10.4, 13.8, 17.2, 20.6, 24, 27.4, 30.8.

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8. Example: Histogram of mpg (2 of 3)

• The upper limits can be taken as 13.8, 17.2, 20.6, 24, 27.4, 30.8, 34.2.

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9. Example: Histogram of mpg (3 of 3)

The following information can be obtained from the histogram.

• The histogram has a single peak within the interval 13.8 and 20.6.

• Majority of cars in the sample has mpg lower than 20.6.

• The mean mpg is around 20.6.

• The range of the mpg is between 13.8 and 34.2.

• The right tail is longer.

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10. Some Remarks on Histogram (1 of 2)

• There should be no space between any two bars.

• The area of a bar represents the relative frequency for the bin. Equivalently, $$\text{area of bar} = \text{relative frequency} = \text{height of bar (density)}\times \text{bar width}.$$

• The vertical axis is called the density scale.

• For continuous variables, unequal bin width may also be used.

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11. Some Remarks on Histogram (2 of 2)

• One convention about the bin width for \(k\) bins is to choose a number with the same or one more decimal place that is greater than \(\frac{\text{range}}{k}\), but no more than \(\frac{\text{range}}{k-1}\) as the bin width.

• To determine the number of bins, there are practical rules. For example, the Rice rule takes the bin number \(k\) as the round up of \(n^{1 / 3}\). The webpage Statistic How To has more information.

• The convenient starting point can be a value smaller but not too much smaller than the minimum.

• The bin width can significantly affect the shape of the histogram. It is better to experiment with different choices.

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Practice: Petal Lengths of Irises

The following data set show the petal length of 20 irises. Construct a frequency table and frequency histogram for the data set using 6 bins. What can you conclude from the histogram?

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Practice: Frequency Table and Histogram

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12. Common Descriptions of Shape Distribution

• Right skewed (or reverse \(J\)-shaped): A right-skewed distribution has a lot of data at lower variable values.

• Left skewed (or \(J\)-shaped): A left skewed distribution has a lot of data at higher variable values with smaller amounts of data at lower variable values.

• Symmetric with a central peak (or bell-shaped): A central peak with a tail in both directions. A bell-shaped distribution has a lot of data in the center with smaller amounts of data tapering off in each direction.

• Uniform: A rectangular shape, the same amount of data for each variable value.

• For examples of left skewed and uniform distributions, please see the example in Dotplot 2 of 2 in Concepts in Statistics

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Practice: Shapes of Distributions

Statistics are used to compare and sometimes identify authors. The following lists shows a simple random sample that compares the letter counts for three authors.

Terry: 7, 9, 3, 3, 3, 4, 1, 3, 2, 2

Davis: 3, 4, 4, 4, 1, 4, 5, 2, 3, 1

Maris: 2, 3, 4, 4, 4, 6, 6, 6, 8, 3

Create a dot plot for each sample and describe the shape of the distribution of each sample.

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Practice: Shapes of Distributions

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13. Centers of a Data Set

• Mean: The mean is the average, this is the quotient of the total sum by the total number.
• Median: The median is a value that separates the data set into the lower half set and the upper half set.
• Mode: The mode is the data value that has the most occurrence in the data set.

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14. Mean and Median for Distributions in Different Shapes

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Practice: Appropriate Measure of Center

A student survey was conducted at a major university. The following histogram shows distribution of alcoholic beverages consumed in a typical week.

1. What is the typical number of drinks a student has during a week?
2. Do the data suggest that drinking is a problem in this university?

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15. Frequency Distribution for Categorical Data

• A frequency distribution for categorical data is a table that displays the possible categories along with the associated frequencies and/or relative frequencies.

• The frequency of a category is the number of occurrences of elements in the category.

• The proportion of a frequency to the size of the population or the sample is also called the relative frequency.

• A relative frequency distribution is a frequency distribution that includes relative frequencies.

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16. Visualization of Categorical Data

• A bar chart consists of bars (rectangles), each represent a category and the area of each bar is constantly proportional to the relative frequency of that category.

• A pie chart is a pie with sectors represents categories and the area of each sector is proportional to the relative frequency of that category.

• Stacked bar char (segmented bar chart) is a bar divided into segments, with each segment representing a category and the area of the segment is proportional to the relative frequency for that category.

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17. Example: Distribution of Majors (1 of 2)

Solution: The relative frequency table is shown below.

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18. Example: Distribution of Majors (2 of 2)

The following are the charts created in Excel.

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Practice: Passengers on Titanic

The following data table summarize passengers on Titanic. Using a chart to describe the data table.

Class	Passengers
1st	325
2nd	285
3rd	706
Crew	885

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Practice: Pie Chart

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19. Frequency Tables

In Excel, to create a frequency table for a data array, we need a bin array. The values in a bin array in Excel are the first \(k-1\) upper bin limits. For example, if the bin array consists of 30, 40, and 50, then the bins will be \([\text{min},30]\), \((30,40]\), \((40, 50]\), \((50, \infty)\).

With a data array and a bin array, the Excel function FREQUENCY(data_array, bins_array) can be used to create a frequency table.

Suppose the data set is in column A and the bin array is in column B.

1. In column C, select a column array of \(k\) cells, then enter =FEQUENCY(
2. select the data values
3. in the formula bar, enter the symbol comma ,
4. select the bin array
5. in the formula bar, enter ).

Hit Enter (Ctrl + Shift + Enter in older versions), you will get a frequency table.

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20. Charts in Excel

Excel has many built-in chart functions. To create a charts,

1. Select the data array/table
2. Under the Insert tab, click on an appropriate chart in the Charts command set.

The appearance of chart can be changed after being created.

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21. Histogram (1 of 2)

1. Select the data

2. On the Insert tab, in the Charts group, from the Insert Statistic Chart dropdown list, select Histogram:

   Note: The histogram contains a special first bin which always contains the smallest number. This is different from many textbooks.

To format the histogram chart is similar to format a Pie chart. For example, you can change bin width from Format Axis.

1. Right-click on the horizontal axis and choose Format Axis in the popup menu:

2. In the Format Axis pane, on the Axis Options tab, you may try different options for bins.

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22. Histogram (2 of 2)

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23. The Analysis ToolPak

Suppose your data set is in Column A in Excel.

• In the cell B1, put the first lower bin limit, which is a number slightly less than the minimum but has more decimal places than the data set.

• Create upper bin limits in column C.

• In Data menu, look for the Data Analysis ToolPak (if not, go to File > Options > Add-ins > Manage Excel Add-ins, check Analysis ToolPak). In the popup windows, find Histogram.

• In the input range, select your data set. In the bin range, select upper bins.

• Check Chart Output and hit OK. You will see the frequency table and histogram in Sheet 2.

• Change the gap between bars. Right-click a bar and choose Format Data Series... and change the Gap Width to 2% or 1%.

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24. Dotplot

• If you have a raw data set, follow the same procedure a creating a histogram but with a bin width equal the same accuracy of the data. For example, if you data set consists of integers, then choose 1 as the bin-width.

• Change the format of bars in the histogram.

  • Right click a bar and select Format Data Series....

  • Find Fill & Line and select both Picture or texture fill and Stack and Scale with.

  • Click the button Oneline... and input dot in search bing and hit enter.

  • Select a picture you like, and you will get a dot-plot.

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Lab Practice: Home State Attending Rates

Describe the distribution of percentage of college students attending college in home states. (To be demonstrated in-class)

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25. Lab Practice: Sleep Deficit and School Start Time

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Lab Practice: Distribution of Random Numbers

Use Excel to complete the following tasks:

1. Create a random sample of 30 two-digit integers.

2. Create a histogram with 6 bins for the sample.

3. Describe the shape of the distribution of the sample of 30 two-digit integers.