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Displaying Quantitative Data Graphically and Describing It Numerically

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Displaying Quantitative Data

- Histogram
- Stem-and-Leaf Plots
- Dotplots
- (Timeplots)

Histograms

- Bins and counts give the distribution of the quantitative data
- Bars touch—data is continuous
- Relative frequency histogram—useful and shows percentages, not counts

Stem-and-Leaf Plot

- Can see each individual data point
- Stem is like bin
- Might need to “split”

- 34779 Key: 3 4 = 34
- 6677789
- 356777777
- 0001
- 99
- 022222

2 577799999 Key: 2 4 = 24

3 44444

- 667789999
- 2333344444

4 577779

Dotplot

- Useful in seeing how many individual data points in bin
- Good for small sets of data
- Not used too often

Describing a Distribution

- Whenever you are describing a distribution you need to describe it by the
- Shape
- Center
- Spread
- Any Unusual points (outliers, gaps)

Shape

- Is the shape?
- Uniform, Symmetric, Skewed

- How many modes (high points)
- Unimodal, bimodal, multimodal

Center and Spread

- How we describe the center and spread of a distribution depends on the shape of the distribution.

Skewed Distribution

- Center: Median
- Spread: Interquartile Range (IQR)
- Both of these are “resistant”
- Both should include units

Skewed Distribution

How to find the IQR

1. Find median

2. Find the median of both halves of data

the lower median is 1st Quartile

the upper median is 3rd Quartile

3. Subtract the two quartile scores

** 1st Quartile = 25th percentile

** 3rd Quartile = 75th percentile

Symmetric Distributions

- Center: Mean
- Spread: Standard Deviation
- Both are not “resistant”
- Both should include units

Properties of standard deviation

- Only use with mean
- If s = 0, there is no spread and all data pieces are same—otherwise s>0 and s gets larger as data pieces get more spread out.
- A few outliers can really change the value of the standard deviation

Other information

- If distribution is symmetric, then mean=median
- If skewed right, mean>median
- If skewed left, mean<median
- Spread of distribution is just as important as the center
- How accurate: one or two decimal points more than original data

Distributions with Outliers

- Really just data that seems unusual
- Formally we compute fences and if data point is outside the fences, we consider it an outlier
- Always use common sense
- Upper fence:
- Lower fence:

Distributions with Outliers

- Tricky situation
- Since outliers affect mean and standard deviation, it is usually better to use median and IQR
- If the mean and median are not similar in value, report the median and IQR
- If the mean and the median are similar in value, report the mean and standard deviation.
- Sometimes (especially if the mean and median are not similar) it is a good idea to report your center and spread with and without the outlier and see what kind of effect removing the outlier has on the distribution.

Boxplots

- Complement histograms by providing more specific information
- Look at histogram and boxplot together
- Most useful when comparing distributions

Boxplots

5-Number Summary: Minimum, 1st Quartile Score, Median, 3rd Quartile Score, Maximum

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