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STAT 2040
Anneke Olthof

STATISTC NOTES Box and Whisker Plots A box and whisker graph is used to display a set of data so that you can easily see where most of the numbers are. For example, suppose you were to catch and measure the length of 13 fish in a lake: A box and whisker plot is based on medians. The first step is to rewrite the data in order, from smallest length to largest: Now find the median of all the numbers. Notice that since there are 13 numbers, the middle one will be the seventh number: This must be the median (middle number) because there are six numbers on each side. The next step is to find the lower median. This is the middle of the lower six numbers. The exact centre is half-way between 8 and 9 ... which would be 8.5 Now find the upper median. This is the middle of the upper six numbers. The exact centre is half- way between 14 and 14 ... which must be 14 Now you are ready to construct the actual box & whisker graph. First you will need to draw an ordinary number line that extends far enough in both directions to include all the numbers in your data: First, locate the main median 12 using a vertical line just above your number line: Now locate the lower median 8.5 and the upper median 14 with similar vertical lines: Next, draw a box using the lower and upper median lines as endpoints: Finally, the whiskers extend out to the data's smallest number 5 and largest number 20: This is a box & whisker plot! B ut what does it mean? What information about the data does this graph give you? Well, it's obvious from the graph that the lengths of the fish were as small as 5 cm, and as long as 20 cm. This gives you the range of the data ... 15. You also know the median, or middle value was 12 cm. Since the medians (three of them) represent the middle points, they split the data into four equal parts. In other words:  one quarter of the data numbers are less than 8.5  one quarter of the data numbers are between 8.5 and 12  one quarter of the data numbers are between 12 and 14  one quarter of the data numbers are greater than 14 The shading below, as an example, shows the quarter of the numbers that are between 12 and 14: Here is a picture of the quarter of the data that is between 8.5 and 12. Notice that the data is more spread out here: This picture is showing where half the data numbers are. Half of all the fish caught had a length between 8.5 and 14 centimetres: Get the idea? See if you can answer the following question: "Below what value is three quarters of the data?" The answer: "Three quarters of the data is below 14." Histograms - breaks the range of values of a variable into classes and displays only the count or percent of the observation What is a Histogram? A histogram is "a representation of a frequency distribution by means of rectangles whose widths represent class intervals and whose areas are proportional to the corresponding frequencies." Sounds complicated . . . but the concept really is pretty simple. We graph groups of numbers according to how often they appear. Thus if we have the set {1,2,2,3,3,3,3,4,4,5,6}, we can graph them like this: This graph is pretty easy to make and gives us some useful data about the set. For example, the graph peaks at 3, which is also the median and the mode of the set. The mean of the set is 3.27—also not far from the peak. The shape of the graph gives us an idea of how the numbers in the set are distributed about the mean: the distribution of this graph is wide compared to size of the peak, indicating that values in the set are only loosely bunched round the mean. How is a Real Histogram Made? The example above is a little too simple. In most real data sets almost all numbers will be unique. Consider the set {3, 11, 12, 19, 22, 23, 24, 25, 27, 29, 35, 36, 37, 45, 49}. A graph which shows how many ones, how many twos, how many threes, etc. would be meaningless. Instead we bin the data into convenient ranges. In this case, with a bin width of 10, we can easily group the data as below. Note: Changing the size of the bin changes the apprearance of the graph and the conclusions you may draw from it. The Shodor histogram activity allows you to change the bin
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