# Describing distributions

Perception of EEO in Division Stem-and-Leaf Plot {see attachment for distributions}

Discussion

The data is based on five questions about EEO with the Columbus Fire Division that were asked on potential job applicants. Each item was a statement about EEO within the Fire Division Response were made on a five point scale from (1) "strongly aprove" (5) "strongly disapprove" The following stem-and leaf plots show the distribution of scores for each person's average on the five EEO items.

Review the results and describe each distribution. How are these distributions different? How are they similar? Is the sample size the same for each?

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## SOLUTION This solution is **FREE** courtesy of BrainMass!

I've pasted this information from a Word document which is attached - the information is identical, so read it in whichever program is easier for you to use. (It's lost a bit of formatting in this version, so I'd suggest you use the attachment instead).

1. Describe each distribution.

Remember that a stem and leaf plot is like a histogram turned on its side. It tells you the frequency of each score; it is another form of a frequency distribution. If you were looking at a histogram, you would eye it up to see if the distribution was normally distributed or not. You can do the same with a stem and leaf. It may help you to mentally rotate the display 90 degrees counter-clockwise (so that it would look like a histogram with the bins (or ranges of scores) on the horizontal axis). Imagine a curve (just like on a graph) that follows the shape of the numbers. Once you've done that, you should be able to eye it up to describe the distribution. Is it normally distributed (ie. is the curve highest in the middle, and does it gradually decrease at an equal rate as the scores spread to each side of the middle)? If not normally distributed, it must be skewed. If it is skewed, you need to figure out in which direction: remember that the direction of the tail, not the hump, is the direction of skew. So if the tail is pointing to the left (ie. negative), the distribution is negatively skewed.

2. Compare the distributions.

To do this, you can simply describe whether they are both normally distributed, and if not, if/how their patterns of skewness differ from one another. You may also want to differentiate between/compare measures of central tendency, such as median (the middle score) and mode (the most frequent score). It is a bit unclear from the question how much detail is expected for this point - generally, the more information you give, the better. You can also talk about extreme points (also known as outliers) in this section and the one above - these are identified on the stem and leaf plot, so you can simply say whether each distribution has them or not for the comparison and descriptions. The number under 'frequency' and beside 'extremes' tells you how many outliers there are in each distribution. You may want to talk about how extreme the extreme scores are (ie. how far they are spread from the next nearest score).

3. Comparing the sample size.

There are two ways you could do this:

a) You could count up the total number of scores from each leaf (the numbers to the right of the decimal points) in the display, and add them up. Remember that the stem and the leaf combine together to tell you a score. For example, if you have a stem of 2 and a leaf of 6, (with a decimal point in between these), that represents a score of 2.6. If you had three leafs of 6 after the 2 and the decimal point, and if each leaf represents one score, that would mean that 3 people scored 2.6. So if you add up the number of leafs in each display, you will know how many scores (ie. how many people) make up that display. This is your sample size. I'll give an example, just in case you're unclear on this:

STEM LEAF NUMBER OF SUBJECTS

1 . 2 1

1 . 44 2

1 . 66666 5

1 . 8 1

2 . 0000 4

Each leaf: 1 case

So your total number subjects in this sample would be 1+2+5+1+4=13.

In your questions, the first data set has each leaf representing 1 case. However, in the second one, each leaf represents 2 cases. This complicates matters a bit. You could count up the number of leafs in each display, multiply that number by two for the second display, and compare them to know if the two samples are the same size. Or, you could use the second (and easier) method, which I would suggest, since there are not always as many leafs as frequency of scores (the number of leafs is rounded down, not up):

b) Look at the frequency column of the displays. This tells you how many people scored in each stem. You'll notice that for the first display, the frequency for each stem is equal to the number of leafs for that stem, but that this is not true for the second display. This is because in the second display, each leaf represents 2 cases. The frequency scores take this into consideration, so the easiest way to do this question is to add up the frequency scores for each display, and compare those to see if the two sample sizes are equal.

Hope this helps!

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