Exploring Data

EMSE 4572/6572: Exploratory Data Analysis

John Paul Helveston

September 30, 2026

Quiz 2

10:00

Write your name on the quiz!

Rules:

• Work alone; no outside help of any kind is allowed.
• No calculators, no notes, no books, no computers, no phones.

Tip of the week:

theme_set()

ggplot(mtcars) +
  geom_point(aes(x = mpg, y = hp))

Default theme

theme_bw(base_size = 20)

Week 6: Exploring Data

1. Exploring Data

2. Data Types

3. Centrality & Variability

4. Visualizing Centrality & Variability

5. Correlation

6. Visualizing Correlation

7. Visualizing Relationships

Week 6: Exploring Data

1. Exploring Data

2. Data Types

3. Centrality & Variability

4. Visualizing Centrality & Variability

5. Correlation

6. Visualizing Correlation

7. Visualizing Relationships

Exploratory Analysis

Goal: Form hypotheses.

Improves quality of questions.

(what we do in THIS class)

Confirmatory Analysis

Goal: Test hypotheses.

Improves quality of answers.

(what you do in a stats class)

Exploratory Analysis

RQ: Do people bike more when the weather is nice?

Confirmatory Analysis

Let’s build a model to predict bike usage based on weather.

Don’t be Icarus

“An approximate answer to the right question is better
than an exact answer to the wrong question.”

— John Tukey

EDA is an iterative process to help you
understand your data and ask better questions

Week 6: Exploring Data

1. Exploring Data

2. Data Types

3. Centrality & Variability

4. Visualizing Centrality & Variability

5. Correlation

6. Visualizing Correlation

7. Visualizing Relationships

24,901

If I walked up to you, and said, “The answer is 24,901,” you would probably be confused. By itself, a number means nothing.

Earth’s circumference at the equator:
24,901

But if I were to tell you that the circumference of the earth at the equator is 24,901 miles, that would mean something.

Earth’s circumference at the equator:
24,901 miles

To be complete and meaningful, quantitative information consists of both quantitative data (the numbers) and categorical data (the labels that tell us what the numbers measure).

Types of Data

Categorical

Subdivide things into groups

• What type?
• Which category?

Numerical

Measure things with numbers

• How many?
• How much?

Categorical (discrete) variables

Nominal

• Order doesn’t matter
• Differ in “name” (nominal) only

e.g. country in TB case data:

#> # A tibble: 6 × 4
#>   country      year  cases population
#>   <chr>       <dbl>  <dbl>      <dbl>
#> 1 Afghanistan  1999    745   19987071
#> 2 Afghanistan  2000   2666   20595360
#> 3 Brazil       1999  37737  172006362
#> 4 Brazil       2000  80488  174504898
#> 5 China        1999 212258 1272915272
#> 6 China        2000 213766 1280428583

Ordinal

• Order matters
• Distance between units not equal

e.g.: Placement 2017 Boston marathon:

#> # A tibble: 6 × 3
#>   Placement `Official Time` Name            
#>       <dbl> <time>          <chr>           
#> 1         1 02:09:37        Kirui, Geoffrey 
#> 2         2 02:09:58        Rupp, Galen     
#> 3         3 02:10:28        Osako, Suguru   
#> 4         4 02:12:08        Biwott, Shadrack
#> 5         5 02:12:35        Chebet, Wilson  
#> 6         6 02:12:45        Abdirahman, Abdi

Numerical data

Interval

• Numerical scale with
  arbitrary starting point
• No “0” point
• Can’t say “x” is double “y”

e.g.: temp in Beaver data

#>   day time  temp activ
#> 1 346  840 36.33     0
#> 2 346  850 36.34     0
#> 3 346  900 36.35     0
#> 4 346  910 36.42     0
#> 5 346  920 36.55     0
#> 6 346  930 36.69     0

Ratio

• Has a “0” point
• Can be described as percentages
• Can say “x” is double “y”

e.g.: height & speed in wildlife impacts

#> # A tibble: 6 × 3
#>   incident_date       height speed
#>   <dttm>               <dbl> <dbl>
#> 1 2018-12-31 00:00:00    700   200
#> 2 2018-12-27 00:00:00    600   145
#> 3 2018-12-23 00:00:00      0   130
#> 4 2018-12-22 00:00:00    500   160
#> 5 2018-12-21 00:00:00    100   150
#> 6 2018-12-18 00:00:00   4500   250

Key Questions

Categorical

Does the order matter?

Yes: Ordinal

No: Nominal

Numerical

Is there a “baseline”?

Yes: Ratio

No: Interval

Be careful of how variables are encoded!

When numbers are categories

• “Dummy coding”: e.g., passedTest = 1 or 0)
• “North”, “South”, “East”, “West” = 1, 2, 3, 4

When ratio data are discrete (i.e. counts)

• Number of eggs in a carton, heart beats per minute, etc.
• Continuous variables measured discretely (e.g. age)

Time

• As ordinal categories: “Jan.”, “Feb.”, “Mar.”, etc.
• As interval scale: “Jan. 1”, “Jan. 2”, “Jan. 3”, etc.
• As ratio scale: “30 sec”, “60 sec”, “70 sec”, etc.

Week 6: Exploring Data

1. Exploring Data

2. Data Types

3. Centrality & Variability

4. Visualizing Centrality & Variability

5. Correlation

6. Visualizing Correlation

7. Visualizing Relationships

Summary Measures:

Single variables: Centrality & Variability

Two variables: Correlation

Centrality (a.k.a. The “Average” Value)

A single number representing the middle of a set of numbers

Mean: \(\frac{\text{Sum of values}}{\text{# of values}}\)

Median: “Middle” value (50% of data above & below)

Mean isn’t always the “best” choice

wildlife_impacts %>%
  filter(!is.na(height)) %>%
  summarise(
    mean = mean(height),
    median = median(height)
  )

#> # A tibble: 1 × 2
#>    mean median
#>   <dbl>  <dbl>
#> 1  984.     50

Percent of data below mean:

#> [1] "73.9%"

On average, at what height do planes hit birds?

On average, where do planes hit birds? Saying ~1000 ft is misleading It’s much more likely to be under 100 ft

Beware the “flaw of averages”

What happened to the statistician that crossed a river with an average depth of 3 feet?

…he drowned

Variability (“Spread”)

Standard deviation: distribution of values relative to the mean

\(s = \sqrt{\frac{\sum_{i=1}^{N}(x_i - \bar{x})^2}{N - 1}}\)

Interquartile range (IQR): \(Q_3 - Q_1\) (middle 50% of data)

Range: max - min

Example: Days to ship

Complaints are coming in about orders shipped from warehouse B, so you collect some data:

daysToShip

#>    order warehouseA warehouseB
#> 1      1          3          1
#> 2      2          3          1
#> 3      3          3          1
#> 4      4          4          3
#> 5      5          4          3
#> 6      6          4          4
#> 7      7          5          5
#> 8      8          5          5
#> 9      9          5          5
#> 10    10          5          6
#> 11    11          5          7
#> 12    12          5         10

Here, averages are misleading:

daysToShip %>%
  gather(warehouse, days, warehouseA:warehouseB) %>%
  group_by(warehouse) %>%
  summarise(
    mean = mean(days),
    median = median(days)
  )

#> # A tibble: 2 × 3
#>   warehouse   mean median
#>   <chr>      <dbl>  <dbl>
#> 1 warehouseA  4.25    4.5
#> 2 warehouseB  4.25    4.5

Example: Days to ship

Complaints are coming in about orders shipped from warehouse B, so you collect some data:

daysToShip

#>    order warehouseA warehouseB
#> 1      1          3          1
#> 2      2          3          1
#> 3      3          3          1
#> 4      4          4          3
#> 5      5          4          3
#> 6      6          4          4
#> 7      7          5          5
#> 8      8          5          5
#> 9      9          5          5
#> 10    10          5          6
#> 11    11          5          7
#> 12    12          5         10

Variability reveals difference in days to ship:

daysToShip %>%
  gather(warehouse, days, warehouseA:warehouseB) %>%
  group_by(warehouse) %>%
  summarise(
    mean = mean(days),
    median = median(days),
    range = max(days) - min(days),
    sd = sd(days)
  )

#> # A tibble: 2 × 5
#>   warehouse   mean median range    sd
#>   <chr>      <dbl>  <dbl> <dbl> <dbl>
#> 1 warehouseA  4.25    4.5     2 0.866
#> 2 warehouseB  4.25    4.5     9 2.70

Example: Days to ship

Interpreting the standard deviation

\(s = \sqrt{\frac{\sum_{i=1}^{N}(x_i - \bar{x})^2}{N - 1}}\)

Outliers

Mean & Standard Deviation are sensitive to outliers

Outliers: \(Q_1 - 1.5 IQR\) or \(Q_3 + 1.5 IQR\)

Extreme values: \(Q_1 - 3 IQR\) or \(Q_3 + 3 IQR\)

Some data:

data1 <- c(3, 3, 4, 5, 5, 6, 6, 7, 8, 9)

• Mean: 5.6
• Standard Deviation: 2.01
• Median: 5.5
• IQR: 2.5

Add just one extreme value:

data2 <- data1
data2[10] <- 20

• Mean: 6.7
• Standard Deviation: 4.95
• Median: 5.5
• IQR: 2.5

Robust statistics for continuous data

(less sensitive to outliers)

Centrality: Use median rather than mean

Variability: Use IQR rather than standard deviation

10:00

Practice with summary measurements

1) Read in the following data sets:

• milk_production.csv
• lotr_words.csv

2) For each variable in each data set, if possible, summarize its

1. Centrality

2. Variability

Week 6: Exploring Data

1. Exploring Data

2. Data Types

3. Centrality & Variability

4. Visualizing Centrality & Variability

5. Correlation

6. Visualizing Correlation

7. Visualizing Relationships

“Visualizing data helps us think”

Stephen Few (2009, pg. 6)

Anscombe’s Quartet

Stephen Few (2009, pg. 6)

Anscombe’s Quartet

The data type determines
how to summarize it

Nominal
(Categorical)

Measures:

• Frequency counts /
  Proportions
  
  
  
  
  

Charts:

• Bars

Ordinal
(Categorical)

Measures:

• Frequency counts /
  Proportions
• Centrality:
  Median, Mode
• Variability: IQR
  
  

Charts:

• Bars

Numerical
(Continuous)

Measures:

• Centrality:
  Mean, median
• Variability: Range, standard deviation, IQR
  
  
  

Charts:

• Histogram
• Boxplot

Summarizing Nominal data

Summarize with counts / percentages

wildlife_impacts %>%
  count(operator, sort = TRUE) %>%
  mutate(p = n / sum(n))

#> # A tibble: 4 × 3
#>   operator               n     p
#>   <chr>              <int> <dbl>
#> 1 SOUTHWEST AIRLINES 17970 0.315
#> 2 UNITED AIRLINES    15116 0.265
#> 3 AMERICAN AIRLINES  14887 0.261
#> 4 DELTA AIR LINES     9005 0.158

Visualize with (usually sorted) bars

wildlife_impacts %>%
  count(operator, sort = TRUE) %>%
  ggplot() +
  geom_col(
    aes(x = n, y = reorder(operator, n)),
    width = 0.7
  ) +
  labs(x = "Count", y = "Operator")

Summarizing Ordinal data

Summarize: Counts / percentages

wildlife_impacts %>%
  count(incident_month, sort = TRUE) %>%
  mutate(p = n / sum(n))

#> # A tibble: 12 × 3
#>    incident_month     n      p
#>             <dbl> <int>  <dbl>
#>  1              9  7980 0.140 
#>  2             10  7754 0.136 
#>  3              8  7104 0.125 
#>  4              5  6161 0.108 
#>  5              7  6133 0.108 
#>  6              6  4541 0.0797
#>  7              4  4490 0.0788
#>  8             11  4191 0.0736
#>  9              3  2678 0.0470
#> 10             12  2303 0.0404
#> 11              1  1951 0.0342
#> 12              2  1692 0.0297

Visualize: Bars

wildlife_impacts %>%
  count(incident_month, sort = TRUE) %>%
  ggplot() +
  geom_col(
    aes(x = as.factor(incident_month), y = n),
    width = 0.7
  ) +
  labs(x = "Incident month")

Summarizing continuous variables

Histograms:

• Skewness
• Number of modes

Boxplots:

• Outliers
• Comparing variables

Histogram: Identify Skewness & # of Modes

Summarise:
Mean, median, sd, range, & IQR:

summary(wildlife_impacts$height)

#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
#>     0.0     0.0    50.0   983.8  1000.0 25000.0   18038

Visualize:
Histogram (identify skewness & modes)

ggplot(wildlife_impacts) +
  geom_histogram(
    aes(x = height),
    bins = 50
  ) +
  labs(x = 'Height (ft)', y = 'Count')

Histogram: Identify Skewness & # of Modes

Height

ggplot(wildlife_impacts) +
  geom_histogram(
    aes(x = height),
    bins = 50
  ) +
  labs(x = 'Height (ft)', y = 'Count')

Speed

ggplot(wildlife_impacts) +
  geom_histogram(
    aes(x = speed),
    bins = 50
  ) +
  labs(x = 'speed (mph)', y = 'Count')

Boxplot: Identify outliers

Height

ggplot(wildlife_impacts) +
  geom_boxplot(aes(x = height)) +
  labs(x = 'Height (ft)', y = NULL)

Speed

ggplot(wildlife_impacts) +
  geom_boxplot(aes(x = speed)) +
  labs(x = 'Speed (mph)', y = NULL)

Histogram

• Skewness
• Modes

Boxplot

• Outliers

15:00

Practicing visual summaries

1. Read in the following data sets:

• faithful.csv
• marathon.csv

2. Summarize the following variables using an appropriate chart (bar chart, histogram, and / or boxplot):

• faithful: eruptions
• faithful: waiting
• marathon: Age
• marathon: State
• marathon: Country
• marathon: `Official Time`

Break!

Stand up, Move around, Stretch!

05:00

Week 6: Exploring Data

1. Exploring Data

2. Data Types

3. Centrality & Variability

4. Visualizing Centrality & Variability

5. Correlation

6. Visualizing Correlation

7. Visualizing Relationships

Correlation: a rather racist origin in eugenics (“well born”)

Sir Francis Galton (1822 - 1911)

• Charles Darwin’s cousin.
• “Father” of eugenics.
• Interested in heredity.

Karl Pearson (1857 - 1936)

• Galton’s (hero-worshiping) protégé.
• Defined correlation equation.
• “Father” of mathematical statistics.

The beautiful irony is that human genetics was also the field that conclusively demonstrated the biological falsity of race.

Galton’s family data

Galton, F. (1886). “Regression towards mediocrity in hereditary stature”. The Journal of the Anthropological Institute of Great Britain and Ireland 15: 246-263.

Galton’s question: Does marriage selection indicate a relationship between the heights of husbands and wives?
(He called this “assortative mating”)

“midparent height” is just a scaled mean:

midparentHeight <- (father + 1.08 * mother) / 2

library(HistData)

galtonScatterplot <- ggplot(GaltonFamilies) +
  geom_point(
    aes(x = midparentHeight, y = childHeight),
    size = 0.5,
    alpha = 0.7
  ) +
  theme_classic() +
  labs(
    x = 'Midparent height (inches)',
    y = 'Child height (inches)'
  )

How do you measure correlation?

Pearson came up with this:

\(r = \frac{\text{Cov}(x, y)}{\text{sd}(x) * \text{sd}(y)}\)

How do you measure correlation?

\(r = \frac{\text{Cov}(x, y)}{\text{sd}(x) * \text{sd}(y)}\)

Assumptions:

1. Variables must be interval or ratio
2. Linear relationship

How do you interpret \(r\)?

\(r = \frac{\text{Cov}(x, y)}{\text{sd}(x) * \text{sd}(y)}\)

Interpretation:

• \(-1 \le r \le 1\)
• Closer to 1 is stronger correlation
• Closer to 0 is weaker correlation

cor(
  x = GaltonFamilies$midparentHeight,
  y = GaltonFamilies$childHeight,
  method = 'pearson'
)

#> [1] 0.3209499

What does \(r\) mean?

• \(\pm 0.1 - 0.3\): Weak
• \(\pm 0.3 - 0.5\): Moderate
• \(\pm 0.5 - 0.8\): Strong
• \(\pm 0.8 - 1.0\): Very strong

Visualizing correlation is…um…easy, right?

guessthecorrelation.com

Click here to vote!

The datasaurus

(More here)

Coefficient of determination: \(r^2\)

Percent of variance in one variable that is explained by the other variable

\(r\)	\(r^2\)
0.1	0.01
0.2	0.04
0.3	0.09
0.4	0.16
0.5	0.25
0.6	0.36
0.7	0.49
0.8	0.64
0.9	0.81
1.0	1.00

You should report both \(r\) and \(r^2\)

Correlation between parent and child height is 0.32, therefore 10% of the variance in the child height is explained by the parent height.

Correlation != Causation

X causes Y

• Training causes improved performance

Y causes X

• Good (bad) performance causes people to train harder (less hard).

Z causes both X & Y

• Commitment and motivation cause increased training and better performance.

Be weary of dual axes!

(They can cause spurious correlations)

Dual axes

Single axis

Outliers

Pearson correlation is highly sensitive to outliers

Spearman’s rank-order correlation

\(r = \frac{\text{Cov}(x, y)}{\text{sd}(x) * \text{sd}(y)}\)

• Separately rank the values of X & Y.
• Use Pearson’s correlation on the ranks instead of the \(x\) & \(y\) values.

Assumptions:

• Variables can be ordinal, interval or ratio
• Relationship must be monotonic (i.e. does not require linearity)

Spearman correlation more robust to outliers

Spearman correlation more robust to outliers

Pearson	Spearman
-0.56	0.53
0.39	0.69
0.94	0.81
0.38	0.76
0.81	0.79
0.31	0.70
0.95	0.81
0.51	0.75
-0.56	0.53

Summary of correlation

• Pearson’s correlation: Described the strength of a linear relationship between two variables that are interval or ratio in nature.
• Spearman’s rank-order correlation: Describes the strength of a monotonic relationship between two variables that are ordinal, interval, or ratio. It is more robust to outliers.
• The coefficient of determination ( \(r^2\) ) describes the amount of variance in one variable that is explained by the other variable.
• Correlation != Causation

R command (hint: add use = "complete.obs" to drop NA values)

pearson <- cor(x, y, method = "pearson", use = "complete.obs")
spearman <- cor(x, y, method = "spearman", use = "complete.obs")

Week 6: Exploring Data

1. Exploring Data

2. Data Types

3. Centrality & Variability

4. Visualizing Centrality & Variability

5. Correlation

6. Visualizing Correlation

7. Visualizing Relationships

Scatterplots: The correlation workhorse

scatterplot <- mtcars %>%
  ggplot() +
  geom_point(
    aes(x = mpg, y = hp),
    size = 2,
    alpha = 0.7
  ) +
  theme_classic(base_size = 20) +
  labs(
    x = 'Fuel economy (mpg)',
    y = 'Engine power (hp)'
  )

scatterplot

Adding a correlation label to a chart

Make the correlation label

corr <- cor(
  mtcars$mpg,
  mtcars$hp,
  method = 'pearson'
)

corrLabel <- paste('r = ', round(corr, 2))

Add label to the chart with annotate()

scatterplot +
  annotate(
    geom = 'text',
    x = 25,
    y = 310,
    label = corrLabel,
    hjust = 0,
    size = 7
  )

Visualize all the correlations: ggcorr()

library('GGally')

mtcars %>%
  ggcorr()

Visualizing correlations: ggcorr()

library('GGally')

mtcars %>%
  ggcorr(
    label = TRUE,
    label_size = 3,
    label_round = 2
  )

Visualizing correlations: ggcorr()

ggcor_mtcars_final <- mtcars %>%
  ggcorr(
    label = TRUE,
    label_size = 3,
    label_round = 2,
    label_color = 'white',
    nbreaks = 5,
    palette = "RdBu"
  )

Pearson

mtcars %>%
  ggcorr(
    label = TRUE,
    label_size = 3,
    label_round = 2,
    method = c("pairwise", "pearson")
  )

Spearman

mtcars %>%
  ggcorr(
    label = TRUE,
    label_size = 3,
    label_round = 2,
    method = c("pairwise", "spearman")
  )

Correlograms: ggpairs()

library('GGally')

mtcars %>%
  select(mpg, cyl, disp, hp, wt) %>%
  ggpairs()

• Look for linear relationships
• View distribution of each variable

Correlograms: ggpairs()

library('GGally')

mtcars %>%
  select(mpg, cyl, disp, hp, wt) %>%
  ggpairs() +
  theme_classic()

• Look for linear relationships
• View distribution of each variable

Your turn

15:00

Using the penguins data frame:

1. Find the two variables with the largest correlation in absolute value (i.e. closest to -1 or 1).

2. Create a scatter plot of those two variables.

3. Add an annotation for the Pearson correlation coefficient.

palmerpenguins library

Artwork by @allison_horst

Simpson’s Paradox: when correlation betrays you

Body mass vs. Bill depth

Body mass vs. Bill depth

Week 6: Exploring Data

1. Exploring Data

2. Data Types

3. Centrality & Variability

4. Visualizing Centrality & Variability

5. Correlation

6. Visualizing Correlation

7. Visualizing Relationships

Visualizing variation

Ask yourself:

• What type of variation occurs within my variables?
• What type of covariation occurs between my variables?

Check out these guides

Two Categorical Variables

Summarize with a table of counts

wildlife_impacts %>%
  count(operator, time_of_day)

#> # A tibble: 20 × 3
#>    operator           time_of_day     n
#>    <chr>              <chr>       <int>
#>  1 AMERICAN AIRLINES  Dawn          458
#>  2 AMERICAN AIRLINES  Day          7809
#>  3 AMERICAN AIRLINES  Dusk          584
#>  4 AMERICAN AIRLINES  Night        3710
#>  5 AMERICAN AIRLINES  <NA>         2326
#>  6 DELTA AIR LINES    Dawn          267
#>  7 DELTA AIR LINES    Day          4846
#>  8 DELTA AIR LINES    Dusk          353
#>  9 DELTA AIR LINES    Night        2090
#> 10 DELTA AIR LINES    <NA>         1449
#> 11 SOUTHWEST AIRLINES Dawn          394
#> 12 SOUTHWEST AIRLINES Day          9109
#> 13 SOUTHWEST AIRLINES Dusk          599
#> 14 SOUTHWEST AIRLINES Night        5425
#> 15 SOUTHWEST AIRLINES <NA>         2443
#> 16 UNITED AIRLINES    Dawn          151
#> 17 UNITED AIRLINES    Day          3359
#> 18 UNITED AIRLINES    Dusk          181
#> 19 UNITED AIRLINES    Night        1510
#> 20 UNITED AIRLINES    <NA>         9915

Two Categorical Variables

Convert to “wide” format with pivot_wider() to make it easier to compare values

wildlife_impacts %>%
  count(operator, time_of_day) %>%
  pivot_wider(names_from = time_of_day, values_from = n)

#> # A tibble: 4 × 6
#>   operator            Dawn   Day  Dusk Night  `NA`
#>   <chr>              <int> <int> <int> <int> <int>
#> 1 AMERICAN AIRLINES    458  7809   584  3710  2326
#> 2 DELTA AIR LINES      267  4846   353  2090  1449
#> 3 SOUTHWEST AIRLINES   394  9109   599  5425  2443
#> 4 UNITED AIRLINES      151  3359   181  1510  9915

Two Categorical Variables

Visualize with bars:
map fill to denote 2nd categorical var

wildlife_impacts %>%
  count(operator, time_of_day) %>%
  ggplot() +
  geom_col(
    aes(
      x = n,
      y = reorder(operator, n),
      fill = reorder(time_of_day, n)
    ),
    width = 0.7,
    position = 'dodge'
  ) +
  theme(legend.position = "bottom") +
  labs(
    fill = "Time of day",
    y = "Airline"
  )

Two Continuous Variables

Visualize with scatterplot - looking for clustering and/or correlational relationship

ggplot(wildlife_impacts) +
  geom_point(
    aes(
      x = speed,
      y = height
    ),
    size = 0.5
  ) +
  labs(
    x = 'Speed (mph)',
    y = 'Height (f)'
  )

One Continuous, One Categorical

Visualize with boxplot

ggplot(wildlife_impacts) +
  geom_boxplot(
    aes(
      x = speed,
      y = operator
    )
  ) +
  labs(
    x = 'Speed (mph)',
    y = 'Airline'
  )

15:00

Practice doing EDA

1. Read in the candy_rankings.csv data sets

2. Preview the data, note the data types and what each variable is.

3. Visualize (at least) three relationships between two variables (guided by a question) using an appropriate chart:

• Bar chart
• Scatterplot
• Boxplot

Reminders:

• You have 4 days until your Mini Project 1 is due (Oct. 4).
• Look at HW 5, due next week