Probability & Statistics Displays of Quantitative Data

Quantitative Data Dealing With a Lot of Numbers… T. Serino Dealing With a Lot of Numbers… Summarizing the data will help us when we look at large sets of quantitative data. Without summaries of the data, it’s hard to grasp what the data tell us. The best thing to do is to make a picture… We can’t use bar charts or pie charts for quantitative data, since those displays are for categorical variables.

Quantitative Data Displays of Quantitative Data Histogram T. Serino Displays of Quantitative Data Histogram Stem-and-Leaf Dot plot

Quantitative Data Displays of Quantitative Data T. Serino Displays of Quantitative Data Box Plots are also used to display quantitative data, but we will not discuss box plots until the next unit.

Quantitative Data Histogram: T. Serino Histogram:  A graph used to display quantitative data.  Similar to a bar graph (with no spaces).  Numbers are grouped into equal width piles called BINS. There are 9 bins in this graph. (one of the bins is empty) If a value falls on the border of two bins, we usually put the value in the bin to the right. If the data were only made up of whole numbers, only numbers 5, 6, 7, 8, and 9 would be stacked in this bin. The data values between 5 and 10 would be stacked in this bin. (including 5 but not including 10) 5 ≤ P < 10 The bin size here is 5. (Each stack is 5 units wide.)

Quantitative Data This histogram could represent the following data: T. Serino This histogram could represent the following data: -18, -16, -12, -12, -11, -11, -7, -6, -5, -5, -4, -4, -4, -4, -3, -3, -3, -3, -3, -2, -2, -2, -2, -1, -1, -1, -1, -1, -1, -1, -1, -1, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 6, 6, 7, 7, 9, 12, 22 That is because the data only has 5 numbers between the numbers 5 and 10. Notice that the bin containing numbers between 5 and 10 is also 5 units high. This bin is empty. The data does not contain any numbers between 15 and 20.

Quantitative Data This histogram could represent the following data: T. Serino This histogram could represent the following data: -18, -16, -12, -12, -11, -11, -7, -6, -5, -5, -4, -4, -4, -4, -3, -3, -3, -3, -3, -2, -2, -2, -2, -1, -1, -1, -1, -1, -1, -1, -1, -1, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 6, 6, 7, 7, 9, 12, 22 Can you tell how tall this bin is? Answer: 21 units high Count the number of data values that are between 0 and 5. Remember, that 0 is included in this bin.

Quantitative Data This histogram could represent the following data: T. Serino This histogram could represent the following data: -18, -16, -12, -12, -11, -11, -7, -6, -5, -5, -4, -4, -4, -4, -3, -3, -3, -3, -3, -2, -2, -2, -2, -1, -1, -1, -1, -1, -1, -1, -1, -1, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 6, 6, 7, 7, 9, 12, 22 Find the height of each of the other bins. The bins are labeled as a range of data. For example, this bin is labeled as 0 ≤ P < 5

Quantitative Data Stem-and-Leaf Displays T. Serino Stem-and-Leaf Displays Usually done by hand therefore used for a relatively small amount of data (approx. 10 to 100 numbers) Leaf Stem -45, -32, -17, -16, -14, -12, -9, -9, -7, -7, -7, -5, -5, -4, -3, -2, 1, 3, 5, 6, 7, 8, 8, 8, 12, 15, 21, 22, 24, 33, 33, 41, 43, 56 If you tilt your head to the right and blur your eyes a little, you can see that a stem-and-leaf plot looks like a histogram. This row represents the numbers 21, 22, and 24.

Quantitative Data Stem-and-Leaf Displays T. Serino Stem-and-Leaf Displays You can also make a double stem-and-leaf plot. (This format is convenient for comparing two related sets of data) Stem Write the data that is represented by the female side of this stem-and-leaf plot. How many males are represented in this plot?

Quantitative Data Dot plots T. Serino Dot plots Similar to a stem-and-leaf plot, but instead of the digits, a dot is placed next to each stem. A double dot plot Dot plots are usually generated by computers and graph a large amount of data.

Quantitative Data Dot plots T. Serino Dot plots The dot plot to the right shows Kentucky Derby winning times, plotting each race as its own dot. You might see a dot plot displayed horizontally or vertically.

Creating Displays of Quantitative Data Display Quantitative T. Serino Creating Displays of Quantitative Data Histograms Stem & Leaf Plots Dot Plots

Display Quantitative Steps to creating a Histogram. T. Serino Steps to creating a Histogram. Put the data in order (from least to greatest) Determine the range of data that needs to be displayed Determine an appropriate bin size Determine how high the largest stack (bin) will be and adjust the frequency scale so that it will fit on the graph. Draw the bars for the Histogram Put a title on the graph and the axes.

Display Quantitative Example: T. Serino Example: The following data lists the test grades of 60 girls and boys in a math class.

Display Quantitative Go back if you haven’t finished yet. T. Serino The first step in creating a display of quantitative data is to put the data in order. Go back if you haven’t finished yet. Take your time! Be careful! Use the space provided in your notes to put the data in order from least to greatest. Your data should look like this.

Display Quantitative T. Serino Next, determine the range of data that needs to be displayed. With a minimum value of 25 and a maximum value of 99, the range of the data is 99 – 25 = 74

Display Quantitative Next, determine and appropriate bin size. T. Serino Next, determine and appropriate bin size. A good rule of thumb is to create a graph that contains at least 7 or 8 bins. Note: The x-axis does not need to show x = 0. The first bin just needs to show the minimum data value. With a range of 74, if we used a bin size of 8, there would be 10 bins. 7 3 5 9 1 4 8 10 2 6

Display Quantitative T. Serino If we used a bin size of 10, there would be 8 bins. 7 3 5 1 4 8 2 6 Because the data is made up of test scores, the appropriate bin size is probably 10. This way, we can clearly see how many students received A’s, B’s, C’s, etc.

Display Quantitative T. Serino Next, we need to determine the largest stack so that we can set the scale on the frequency axis. Because our biggest stack (bin) will be 17 units high, the frequency axis has to range from zero to at least 17. (Remember, that zero must be included in the frequency axis.) There are 10 values that are less than 100 and greater than or equal to 90. 90 ≤ s < 100 There are 14 values that are less than 90 and greater than or equal to 80. 80 ≤ s < 90 There are 17 values that are less than 80 and greater than or equal to 70. 70 ≤ s < 80 This is our largest stack!

Display Quantitative T. Serino The following graph has 10 tick marks (boxes) available for each axis. To fit 17 units on the vertical (frequency) axis…. 2 4 6 8 10 12 14 16 18 20 Scale = 2 Using a scale of 10 on the horizontal scale will allow us all 8 bins that we need. 10 20 30 40 50 60 70 80 90 100

Display Quantitative Next, let’s draw the graph. T. Serino Next, let’s draw the graph. Math Test Grades 2 4 6 8 10 12 14 16 18 20 One number in the 20 ≤ s < 30 bin The 30 ≤ s < 40 bin is empty. One number in the 40 ≤ s < 50 bin Seven numbers in the 50 ≤ s < 60 bin Ten numbers in the 60 ≤ s < 70 bin Seventeen numbers in the 70 ≤ s < 80 bin Fourteen numbers in the 80 ≤ s < 90 bin Number of Students Ten numbers in the 90 ≤ s < 100 bin Don’t forget the labels! 10 20 30 40 50 60 70 80 90 100 Scores

Display Quantitative T. Serino Stem & Leaf Plots and Dot Plots are drawn almost exactly as Histograms are drawn. If you blur your eyes just a bit while looking at Stem & Leaf plots and dot plots, you can see that they are really the same type of display.

Display Quantitative T. Serino The following are the ages of the female members of an orchestra. Once we organize the data into ascending order, we can easily draw a stem & leaf plot. For this data, the 10’s digit would be the proper stem. 41, 55, 60, 72, 21, 15, 18, 29, 42, 61, 29, 26, 46, 48 15, 18, 21, 26, 29, 29, 41, 42, 46, 48, 55, 60, 61, 72 A double stem & leaf plot would be a nice way to compare male and female members of the orchestra. Ages of male members: 10, 12, 15, 25, 21, 31, 33, 35, 35, 35, 42, 45, 51, 51, 56, 58, 59, 60, 65, 65, 66, 70, 70, 71, 71, 72

Display Quantitative T. Serino Female members: 15, 18, 21, 26, 29, 29, 41, 42, 46, 48, 55, 60, 61, 72 The stem & leaf plot drawn in the previous example, could also be drawn as a dot plot. Male members: 10, 12, 15, 25, 21, 31, 33, 35, 35, 35, 42, 45, 51, 51, 56, 58, 59, 60, 65, 65, 66, 70, 70, 71, 71, 72 For the dot plot, The “stem” would display the scale in the same way as a histogram. 10 20 30 40 50 60 70 The “leafs” would be displayed by dots instead of by specific numbers.

Try this. T. Serino Display the following data using a histogram with bin size 5. (use the grid provided in your notes)

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