Jan 12, 2026

IT 223 -- Jan 12, 2026

Review Problems

Which R functions do you know? Answer:

print  cat  c  seq  mean  median  IQR  
summary  quantile  stem  hist  boxplot

Compute Q1, Q3, IQR for this sorted dataset, using the Tukey's Hinges method.
```
13  27  35  39  53  121  983 
```
Answer: Q2 is the middle number 39, Q1 is the median of the bottom half of the data 31, Q3 is the median of the top half of the data 87. Recall that if n, the number of observations, is odd, that the middle number is included in both the bottom half and the top half, so
    Q1 = (27 + 35) / 2 = 31
and
    Q3 = (53 + 121) / 2 = 87.
Also
    IQR = Q3 - Q1 = 87 - 31 = 56.
What information is contained in a boxplot?
Answer: Q0, Q1, Q2, Q3, Q4, and any outliers, which are below the lower inner fence or above the upper inner fence.
What does the term inner fence mean?
Ans: The two inner fences are located at IF₁ = Q1 - 1.5 * IQR and IF₂ = Q3 + 1.5 * IQR. Outliers are points which are less than IF₁ or greater than IF₂.
What does it mean for two quantities to be linearly proportional? Give some examples. Answer:
Example 1: Baking Bread. Define these quantities in cups:
s₁ = amount of sugar for original recipe
s₂ = amount of sugar for doubled recipe
f₁ = amount of flour for original recipe
r₂ = amount of flour for doubled recipe
The amount of sugar and flour are proportional in the original and doubled recipes, which means
s₂ / s₁ = f₂ / f₂.
Example 2: Show that time and distance traveled are proportional when driving at a constant speed.
Answer: show that t₂ / t₁ = d₂ / d₁.
When driving at a constant speed, distance = time * velocity, so
t₂ / t₁ = d₂ / d₁
t₂ / t₁ = (v * t₂) / (v * t₁)
The v in the numerator cancels with the v in the denominator, which makes the two sides of the equation equal.
Given two rectangles that are the same height, show that the areas of the rectangles are proportional to their heights, using the fact that area = base * height or A = b * h for a rectangle. Answer:
A₂ / A₁ = b₂ / b₁ = (h * b₂) / (h * b₁) = b₂ / b₁,
where we cancel the h in the numerator with the h in the denominator.
Compute the medians of the histograms from tables b and c for Problem 1 in More Practice Problems at the bottom of the Jan 7 Notes. The answer is shown at the end of this problem.
Compute Q1 (25th percentile) and Q3 (75th percentile) of Table c from the previous problem.
Answer: The calculations for Q1 and Q3.
Draw the histogram without bar lines of
1. (i) the incomes of all persons in the U. S.
  Answer: A histogram skewed to the right with a peak at about 65 thousand, but with a long right tail that extends all the way past 1 billion
2. (ii) the GPAs of all students at DePaul. Answer: A bell-shaped histogram with peak around 3.0. There may be a secondary peak around 2.0, representing those students that have just come off of academic probation. There are very few students with GPA < 1.0 or GPA close to 4.0. Most GPAs will be between 30 and 3.5. The height of the histogram can only be nonzero in the range from 0 to 4.
3. the number of years of schooling of all persons in the U. S.
  Answer: A bell-shaped peak around 12 years (most people finish highschool, less people attend college). There may be a secondary peak close to 16 years which represents the people that complete college.
4. the IQs of all persons in the U.S. Answer: A bell-shaped curve with center at 100 and spread 15. IQ scores are scaled to look like this.

NIST 10 Gram Prototype Weight Weighings Dataset

One of the responsabilities of the National Institute of Standards and Technology (NIST) is maintaining standards for accurate measurements, including weights measurements.
Look at the AI Overview for the question "what does NIST do?"
The NIST 10 Gram Prototype Weight Weighings Dataset is the second dataset posted on the R Datasets Page. It consists of weight measurements of a 10 gram prototype weight once a week for 100 weeks (almost two years). This prototype is weighed with a scale that measures weights to the nearest microgram (10^-6 gram).
To create a vector of these 100 weighings for R to use, first create a folder named workspace to use for the calculations. This folder should be created in the root folder C:/ for windows. Then copy the nist10.txt database into your new workspace folder. If you wish, you can use R to create this new workspace folder:
```
setwd("C:/")
dir.create("workspace")
```
I need a volunteer that uses a Mac to help me write a short description of how to create the folder named workspace in the home directory. On a Mac, you should be able to use to create the workspace folder like this:
```
setwd("~")
dir.create("workspace")
```

Here is an R script that creates the vector of these weights named w to be analyzed using R. Note that an R comment line is marked with # at the beginning of the line.

# Obtain the current working directory.
getwd( )

# Set the current working directory to C:/workspace.
setwd("C:/workspace")

# When using unix, set the current working directory 
# to ~/workspace where ~ is the home directory.
# setwd("~/workspace")

# Create a dataframe named weightsDf from the file nist10.txt
# This dataframe will contain the columns 
weightsDf <- read.csv("nist-10.txt")

# Extract a vector named w from the Weight column
# of the data frame
w <- weightsDf$Weight

# Print the weight vector
print(w)

When you copy and paste this script into R and press Enter, the w vector is printed:

  [1] 9.999591 9.999600 9.999594 9.999601 9.999598 9.999594 9.999599 9.999597
  [9] 9.999599 9.999597 9.999602 9.999597 9.999593 9.999598 9.999599 9.999601
 [17] 9.999600 9.999599 9.999595 9.999598 9.999592 9.999601 9.999601 9.999598
 [25] 9.999601 9.999603 9.999593 9.999599 9.999601 9.999599 9.999597 9.999600
 [33] 9.999590 9.999599 9.999593 9.999577 9.999594 9.999594 9.999598 9.999595
 [41] 9.999595 9.999591 9.999601 9.999598 9.999593 9.999594 9.999587 9.999591
 [49] 9.999596 9.999598 9.999596 9.999594 9.999593 9.999595 9.999589 9.999590
 [57] 9.999590 9.999590 9.999599 9.999598 9.999596 9.999595 9.999608 9.999593
 [65] 9.999594 9.999596 9.999597 9.999592 9.999596 9.999593 9.999588 9.999594
 [73] 9.999591 9.999600 9.999592 9.999596 9.999599 9.999596 9.999592 9.999594
 [81] 9.999592 9.999594 9.999599 9.999588 9.999607 9.999563 9.999582 9.999585
 [89] 9.999596 9.999599 9.999599 9.999593 9.999588 9.999625 9.999591 9.999594
 [97] 9.999602 9.999594 9.999597 9.999596

We can now analyze this w vector like we did last week with the Celsius temperatures in Review Exercise 10 of the Jan 7 Lecture Notes, and also like you are analyzing the hypothetical exam scores for Project 1. Use these R functions to perform the analysis:
```
median  IQR  quantile  hist  boxplot  mean
```

Descriptive Statistics

If a histogram is bell-shaped or normal, it can be described by its center μ and spread σ:
Many datasets that occur in real life are approximately normally distributed, for example: heights of men, heights of women, weights of newborn human babies, blood pressures of humans, IQ scores, velocities of molecules in an ideal gas, age of NFL players at retirement.
For a bell-shaped histogram, if the sample size n is large, the statistics x (sample mean) and SD+ (sample standard deviation) are good estimates of μ and σ.
x and SD+ form a parsimonious description of a bell-shaped histogram.
Estimates of the Center: Mean Median Trimmed Mean
Estimators of the Spread: SD SD⁺ MAD