Introduction to Probability Models

Lecture 35

Qi Wang, Department of Statistics

Nov 15, 2017

Measures of Spread

  • Range
  • Variance
  • Standard deviation
  • $p_{th}$ percentile
  • Interquartiles Range(IQR)


  • Range = max - min


Variance: based on the difference between each observation and the mean

  • Population variance: $$\sigma^2 = \frac{\sum(x_i - \mu)^2}{N}$$
  • Sample variance: $$s^2 = \frac{\sum(x_i - \bar{x})^2}{n - 1}$$

Standard Deviation

Standard deviation: most commonly used for measuring how far observation are from the mean

  • Population version: $$\sigma = \sqrt{\sigma^2}$$
  • Sample version: $$s = \sqrt{s^2}$$

$p_{th}$ percentile

$p_{th}$ percentile: value such that p% of the observation fall at or below it

  • Median: $M = 50_{th}$ percentile
  • First quartile: $Q_1 = 25_{th}$ percentile
  • Third quartile: $Q_3 = 75_{th}$ percentile

How to Find a Percentile for Data

  1. Order the data in increasing order
  2. Calculate $i=\frac{np}{100}$, where $n$ is the sample size, $p$ is the percentile
    • If $i$ is not an integer, round $i$ up to the next integer. Then take the $i_{th}$ value
    • If $i$ is an integer, take an average of the $i_{th}$ and $(i + 1)_{th}$ values

Example: -20, 1, 23, 25, 32.5, 33, 67

Interquartiles Range(IQR)

  • IQR = $Q_3 - Q_1$
  • Outliers: an observation is said to be a suspected outlier if it is $$> Q_3 + 1.5*IQR$$ OR $$< Q_1 - 1.5 * IQR$$


Boxplot is a graphic depiction of the 5 number summary

  1. Draw a horizontal or vertical axis that is evenly spaced and well-labeled(make sure it covers the full range of the data)
  2. Locate $Q_1$ and $Q_3$. There are the "ends" of your box. Draw the box.
  3. With the box, locate the Median and mark it
  4. Locate and mark the Minimum and Maximum. Extend a line("whisker") from each end of the box to the Max or Min

Modified Boxplot

step 1, 2, 3 are the same. BUT we indicate the outliers with a $o$ or a $\star$. Then draw the line from the ends of the box ot the highest or lowest data point that is NOT an outlier. Most software generate boxplots are modified boxplots.