Interpreting Categorical and Quantitative Data

Center, Shape, Spread, and unusual occurrences When describing graphs of data, we use central tendencies (mean, mode, median), shape (unimodal, skewed, uniform, symmetrical etc.), spread (range, interquartile range, standard deviation) and unusual occurrences (outliers, clusters etc.). These items also allow us to compare graphs of the same type.

What strikes you as the most distinctive difference among the distributions of exam scores in classes A, B, & C ?

1. Center discuss where the middle of the data falls three types of central tendency mean, median, & mode

What strikes you as the most distinctive difference among the distributions of scores in classes D, E, & F?

2. Spread discuss how spread out the data is refers to the variability of the data Range, standard deviation, IQR

What strikes you as the most distinctive difference among the distributions of exam scores in classes G, H, & I ?

3. Shape refers to the overall shape of the distribution symmetrical, uniform, skewed, or bimodal

What strikes you as the most distinctive difference among the distributions of exam scores in class K ?

4. Unusual occurrences outliers - value that lies away from the rest of the data gaps clusters anything else unusual

Example Describe the following distribution

Start with Central Tendencies Describe the following distribution of test scores. Yes, there was a bonus question. You are going to have to look at the frequencies. The first bar has approximately 3 The second bar has approximately 10 The Third bar has approximately 13 The Fourth bar has approximately 28 The last bar has approximately 20 So there is a total of approximately 74 items. So the middle term or Median would be the average Of the 36 th and 37 th term. What bar is that in?

Example The median is in the 77.5 to 92.5 range. A mean could be found by taking the midpoint of each bar, multiplying it by its frequency and then dividing by 74

Discuss Spread The easiest would be the range, max – min – 32.5 =

Shape If we drew in a curve, we would say this data was skewed left. It is also unimodal

Unusual Occurrences This particular graph does not have unusual occurrences

Put it all together The distribution of the test data shows a median in the 77.5 to the 92.5 data set. The test grades ranged from 32.5 to and the grades were skewed left or to the lower numbers. Students would want a test skewed to the lower numbers.

Example By construction of a boxplot, we easily get the Median for our center, range and interquartile range, as well as the shape of the data. A modified box plot would include outliers, but outliers can be found on this traditional one as well. To decide if there is a potential outlier, multiply 1.5 (IQR) and subtract it from Q1 or add it to Q3. If your “whisker” extends beyond this point, you have at least one outlier.