Summarizing Numerical Data

STAT 20: Introduction to Probability and Statistics

Adapted by Gaston Sanchez

Agenda

• Announcements
• Notes Recap
• Practice Concept Questions
• Coding Activity: Graphing Numerical Data
• Break
• Worksheet “Summarizing Numerical Data” (part of WSP-3)
• Lab 2.2 (time permitting)

Announcements

• RQ: Grammar of Graphics due Thursday at 11:59pm

• Lab 2: Class Survey (both parts) due Tuesday at 8am

  • Lab 2.1: Group submission
  • Lab 2.2: Individual submission

Notes Recap

Summarizing Distributions of Data using:

• Graphics
• Numerical Summaries

Graphics

You can construct a statistical graphic to show the shape, which you can describe in terms of modality and skew

• Dot plot
• Histogram
• Density plot
• Violin plot
• Box plot

Measures of Center

You can calculate a measure of center to convey a sense of a typical (representative) observation

• Mean
• Median
• Mode

Measures of Spread

And you can calculate a measure of spread (i.e. scatter, dispersion, variation) to capture how much variability there is in the data

• Range
• Inter Quartile Range (IQR)
• Mean Absolute Deviation (MAD)
• Sample Variance (Var)
• Sample Standard Deviation (SD)

Typical value?

\[ 6 \quad 7 \quad 7 \quad 7 \quad 8 \quad 8 \quad 9 \quad 9 \quad 10 \quad 11 \quad 11\]

• sample size ( \(n\) ): 11
• mean ( \(\bar{x}\) ): 8.45
• median: 8
• mode: 7

How can we express the variability in this data set using a single number?

\[ 6 \quad 7 \quad 7 \quad 7 \quad 8 \quad 8 \quad 9 \quad 9 \quad 10 \quad 11 \quad 11\]

\[ {\Large 6} \quad 7 \quad 7 \quad 7 \quad 8 \quad 8 \quad 9 \quad 9 \quad 10 \quad 11 \quad {\Large 11}\]

The Range

\[\textrm{range:} \quad max - min\]

\[ 11 - 6 = 5\]

Characteristics

• Very sensitive to extreme values!

\[ 6 \quad 7 \quad {\Large 7 \quad 7} \quad 8 \quad {\large 8} \quad 9 \quad {\Large 9 \quad 10} \quad 11 \quad 11\]

The Inner Quartile Range (IQR)

The difference between the 3rd quartile, \(Q_3\), and the 1st quartile, \(Q_1\) (i.e. the middle 50% of the data)

\[\textrm{IQR:} \quad Q_3 - Q_1\]

\[ 9.5 - 7 = 2.5 \]

Characteristics

• Robust to outliers
• Used to set the width of the box in a boxplot

\[ 6 \quad 7 \quad 7 \quad 7 \quad 8 \quad 8 \quad 9 \quad 9 \quad 10 \quad 11 \quad 11\]

Mean Absolute Deviation

Take the differences from each observation, \(x_i\), to the sample mean, \(\bar{x}\), take their absolute values, add them up, and divide by \(n\). Simply put, this is the average distance from the mean.

\[MAD: \quad \frac{1}{n}\sum_{i = 1}^n |x_i - \bar{x}| \]

\[ MAD = 1.4 \]

Characteristics

• Incorporates information from all observations
• Robust to extreme values

\[ 6 \quad 7 \quad 7 \quad 7 \quad 8 \quad 8 \quad 9 \quad 9 \quad 10 \quad 11 \quad 11\]

Sample Variance

Take the differences from each observation, \(x_i\), to the sample mean, \(\bar{x}\), square them, add them up, and divide by \(n - 1\) .

\[s^2: \quad \frac{1}{n - 1}\sum_{i = 1}^n (x_i - \bar{x})^2 \]

\[ s^2 = 2.87 \]

Characteristics

• Incorporates information from all observations

• Moderately sensitive to extreme values

• Measured in squared units!

\[ 6 \quad 7 \quad 7 \quad 7 \quad 8 \quad 8 \quad 9 \quad 9 \quad 10 \quad 11 \quad 11\]

Sample Standard Deviation

Take the differences from each observation, \(x_i\), to the sample mean, \(\bar{x}\), square them, add them up, divide by \(n - 1\), then take the square root.

\[ S: \sqrt{\frac{1}{n - 1}\sum_{i = 1}^n (x_i - \bar{x})^2} \]

\[ s = 1.70 \]

Characteristics

• Incorporates info from all observations

• Moderately sensitive to extreme values

• Measured in units of the original data

Practice Concept Questions

15:00

Introducing ggplot2

Demo

tinyurl.com/ybhwtrr9

Coding Activity: Graphing Numerical Data

25:00

Break

05:00

Worksheet: Summarizing Numerical Data

30:00

Lab 2.2 Time Permitting

End of Lecture