MAS 261 - Lecture 3

Housekeeping

• Today’s plan 📋

  • Review Question about Measures of Central Tendancy

  • A few minutes for R Questions 🪄

  • Measures of Variability

    • How do we determine variability (spread)

    • Sample and Population measures

  • Examining Data Variability Visually

    • Boxplots

  • In-class Exercises

R and RStudio

• In this course we will use R and RStudio to understand statistical concepts.

• You will access R and RStudio through Posit Cloud.

  • Sign up for a Free Posit Cloud Account

• I will post R/RStudio files on Posit Cloud that you can access in provided links.

• I will also provide demo videos that show how to access files and complete exercises.

• NOTE: The free Posit Cloud account is limited to 25 hours per month.

  • I will demo how to download completed work so that you can use this allotment efficiently.

  • For those who want to go further with R/RStudio:

    • After Test 1, I will provide videos on how to download the software (R/RStudio/Quarto) and lecture files to your computer.

💥Lecture 3 In-class Exercises - Q1 💥

Session ID: MAS261f24

In Lecture 2 we discussed the measures of central tendency, the mean, median, and mode.

Which of these measures can ALWAYS be found and is good measure of central tendancy regardless of whether or not the data has extreme values (high or low extremes) referred to as outliers?

A. Mode

B. Mean

C. Median

Measures of Variability and Deviations

• Measures Central Tendancy such as MEAN and MEDIAN indicate where the center of the data is located.

• Measures of Variability indicate how spread out the data observations are.

• The building block used to calculate many measures variability is called a DEVIATION.

  • DEVIATION: How far an observation is above or below the data mean.
  • DEVIATION (of an observation) - Observation - mean

• Demonstration of calculating deviations (2 options shown):

  • Notice that deviations are both positive (above the mean) and negative (below the mean)

🤯 R Demo of Calculating Deviations

• Notice that some deviations are positive and others are negative.
• A positive deviation indicate an observation is above the mean.
• A negative deviation indicates an observation is below the mean.

Total Sums of Squares (TSS)

• If we sum all of these deviations we would get zero

• We could sum absolute values of the deviations, but the underlying math (Calculus) wouldn’t work well.).

• INSTEAD we sum the SQUARED deviations and take the square root at the end of our calculations.

Variance (Var)

• R command for Variance: var

• \(Var = \frac{TSS}{n-1}\) where n is the number of observations in variable.

• \(TSS = Var \times (n-1)\)

• In HW 2 you will calculate TSS from Variance using this simple equation.

var(my_cars$mpg)               # variance of mpg

[1] 36.3241

var(my_cars$mpg)*31            # tss of mpg

[1] 1126.047

Standard Deviation

• \(SD = \sqrt{Var} = \sqrt{\frac{TSS}{n-1}}\)

• \(Var = SD^2\)

• R command: sd

• Sample Standard Deviation is the measure we use most often in MAS 261.

sd(my_cars$mpg)             # std deviation calculation

[1] 6.026948

(sd(my_cars$mpg))^2         # calculating variance from std. dev

[1] 36.3241

Coefficient of Variation (CV) and Range

• \(CV = \frac{SD}{\overline{X}}\), SD divided by sample mean, xbar

  • In R sd()/mean() where the dataset and variable are specified in parentheses.

• Range = data maximum minus date min

• In R, the range command outputs the data minimum and maximum

sd(my_cars$mpg)/mean(my_cars$mpg)       # cv calculation

[1] 0.2999881

range(my_cars$mpg)

[1] 10.4 33.9

Relationships between Measures of Variability

• For TSS, Variance, Standard Deviation, and CV, deviations are the building blocks

• \(Var = \frac{TSS}{n-1}\) and \(TSS = Var \times (n-1)\)

• \(SD = \sqrt{Var}\) and \(Var = SD^2\)

• \(CV = \frac{SD}{\overline{X}}\) and \(SD = CV \times \overline{X}\)

  • Recall that \(\overline{X}\) is the symbol the sample mean.

• The two measures we will use most often are Standard Deviation (SD) and CV.

• Later in the course we will also use Variance.

Variability Calculations in Excel

• All of the calculations shown today (and in lecture 2) can be done in Excel

• For large datasets, however, it makes sense to to use R or another coding language

• Starting with Lecture 4, Excel will become less practical to use.

• The R dataset, mtcars has been exported to Excel and can be accessed here.

Sample and Population measures

• In R, it is assumed that you are calculating measures of variability from a sample

• Sample STATISTICS

• In Excel, you have to specify by using =stdev.s and =var.s for sample data

• We will USUALLY not use the population calculations in this course,

  • If you use Excel, DO NOT USE =stdev.p and =var.p unless specified.

• Be aware that THEY ARE SLIGHTLY DIFFERENT

💥Lecture 3 In-class Exercises - Q2 💥

Session ID: MAS261f24

For these exercises we will use the R starwars dataset. You can use Excel or R, but I recommend R.

• The R dataset, mtcars has been exported to Excel and can be accessed here.
  • This file Shows the Excel formulas for all the measures in Lectures 2 and 3.
• The R dataset starwars has been Exported to Excel and can be accessed here.

my_starwars <- starwars |> 
  filter(!is.na(height)) |> 
  select(name, height, species)

💥Lecture 3 In-class Exercises - Q3 💥

Session ID: MAS261f24

Recall that the deviation for an individual observation is observation minus mean.

What is the deviation from the mean for the height of R2-D2? Round answer to closest whole number and include negative sign if needed.

Height for R2-D2 is 96 cm. Use this code to answer this question.

96 - mean(my_starwars$height)

💥Lecture 3 In-class Exercises - Q4 💥

Session ID: MAS261f24

Recall that \(SST = Variance\times(n-1)\) and \(Variance = \frac{SST}{n-1}\)

What is the SST, the total sum of the squared deviations for the height variable in the StarWars dataset? Round answer to closest whole number.

To find this answer, find the variance of height and multiply it by the sample size minus 1

var(my_starwars$height)*80

💥Lecture 3 In-class Exercises - Q4 💥

Session ID: MAS261f24

Recall that \(CV = \frac{SD}{\overline{X}}\), Standard Deviation divided by sample mean, \(\overline{X}\).

CV is useful for comparing data with different units, e.g. centimeter data to inches, or US dollars to British pounds because the units are factored out.

What is the CV, the coefficient of variation for the height variable in the StarWars dataset? Round answer to two decimal places

sd(my_starwars$height)/mean(my_starwars$height)

Min, Max, Median, Quartiles, and Percentiles

There are values in the data that are used to understand it’s variability and distribution

We’ve already discussed these values:

• Minimum: lowest value in the data

• Maximum: highest value in the data

• Range: Maximum minus Minimum

• Median: The middle value or average of two middle values

  • Also referred to as the 50th percentile or the 2nd Quartile
  • 50% (two quarters) of the observations are below this value
  • 50% (two quarters) of the observations are above this value

• 25th percentile, also called the 1st Quartile: 25% of the data is below this value
• 75th percentile, also called the 3rd Quartile: 75% of the data is below this value

Five Number Summary

• Minimum, 25th Percentile, Median, 75th Percentile, Maximum

• In R, we use the command summary to calculate these values and also get the mean as a bonus.

summary(my_cars$mpg)                                                                            

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  10.40   15.43   19.20   20.09   22.80   33.90 

Visualizing Data

• Measures of Central Tendancy
• Measures of Variability
• Quartiles and Percentiles (Percentiles are also called Quantiles)
• Extreme Values also referred to as Outliers
• Comparing different categories

Five Number Summary and Boxplots

• Today, we filter this dataset to create realtor1 with:

  • two states, Maine and Vermont.

  • houses with prices of $1.2 Million or less.

• We also make 2 separate datasets:

  • realtor_VT includes data for Vermont only

  • realtor_ME includes data for Maine only

Five Number Summary for Each State

• Minimum, Q1, Median (Q2), Q3, Maximum and Mean

summary(realtor_VT$price)                                                  

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  10000   89000  200000  250091  343750 1200000 

summary(realtor_ME$price)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  14999  174500  349900  391567  569000 1200000 

• In each state summary, the mean is substantially higher than the median.

• Summary values for Vermont are shown on a boxplot (next slide).

• In HW 2, you will annotate annotate a boxplot of the data used for that assignment about commute times.

• Excel is NOT RECOMMENDED for finding these values!

Boxplot Annotated with Five Number Summary

Five Number summary for Vermont:

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  10000   89000  200000  250091  343750 1200000 

Key Points from Today

• Measures of Measures of Variability

  • Deviation is ‘building block’

    • Deviation = Observation - Mean

    • TSS, Variance, Standard Deviation and CV are all related

    • Range is Maximum minus Minimum

    • Quartiles, Q1, Q2, Q3 also help describe variability

    • summary command shows Min., Q1, Q2, Q3, Max. and Mean

    • Boxplot shows Min., Q1, Q2, Q3, Max.

    • Boxplot shows additional information (Lecture 4)