By the end of today you can

• explain the place of exploratory analysis in research;
• choose a summary that suits the level and shape of a variable;
• summarise centre and spread, and explain the standard deviation;
• describe shape, and check it against the normal distribution;
• find unusual values, and decide what to do with them;
• describe how much data is missing, and why;
• interpret correlations, and why correlation is not causation;
• describe a sample in a thesis.

Description is not a formality

• the shape of a variable decides which tests are appropriate;
• an unusual value can dominate an analysis;
• missing data can quietly change who the results are about.

Descriptive statistics are the first results in almost every thesis.

Three kinds of analysis

Exploration comes first

flowchart TB
  subgraph R1[" "]
    direction LR
    A[Question] --> B[Import] --> C[Clean]
  end
  subgraph R2[" "]
    direction LR
    D[Explore] --> E[Test or predict] --> F[Report]
  end
  R1 --> R2

Return to exploration whenever a later result surprises you.

Three measures of the centre

\[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i \]

One heavy user moves the mean

caffeine <- c(0, 120, 150, 180, 900)
mean(caffeine)
#> [1] 270
median(caffeine)
#> [1] 150

The mean is higher than four of the five students.

The mean is sensitive to extreme values; the median is not.

Mean and median in the study

first_sem |>
  summarise(mean_sleep      = mean(sleep_hours, na.rm = TRUE),
            median_sleep    = median(sleep_hours, na.rm = TRUE),
            mean_caffeine   = mean(caffeine_mg, na.rm = TRUE),
            median_caffeine = median(caffeine_mg, na.rm = TRUE))

Why caffeine’s mean is higher

A minority of heavy users pull the mean (solid) above the median (dashed).

Choosing a summary

The level of measurement comes first, then the shape.

Same average, different stories

group_a <- c(6.0, 6.5, 6.5, 7.0, 6.5)
group_b <- c(4.0, 8.5, 5.0, 9.0, 6.0)

Both average 6.5 hours.

In group A everyone sleeps about the same; group B ranges from 4 to 9 hours.

A mean without a measure of spread hides half the story.

Deviations from the mean

Each copper line is a deviation, \(x_i - \bar{x}\). The standard deviation summarises them.

The standard deviation

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

• deviations add to zero, so they are squared first;
• the average squared deviation is the variance;
• its square root returns to the original units.

Dividing by \(n - 1\) gives a better estimate from a sample.

The SD is roughly the typical distance

sd(group_a)                          #> 0.35
sd(group_b)                          #> 2.18
mean(abs(group_b - mean(group_b)))   #> 1.80

Group B’s SD is close to the average length of the copper lines.

It is a little larger, because squaring weights the longest lines more.

The interquartile range

quantile(first_sem$caffeine_mg, c(0.25, 0.5, 0.75), na.rm = TRUE)
#> 25% 50% 75%
#> 105 160 240
IQR(first_sem$caffeine_mg, na.rm = TRUE)
#> [1] 135

The spread of the middle half: \(Q_3 - Q_1\). It ignores the extremes, like the median.

A box plot’s box is exactly the IQR.

Pairs that belong together

Sleep: SD ≈ 1 hours, so a typical student sleeps within about an hour of 6.5.

The shape of a distribution

Sleep is symmetrical, caffeine is not

scales = "free" gives each panel its own axes: hours and milligrams differ greatly.

Skewness puts a number on it

library(psych)
skew(first_sem$sleep_hours, na.rm = TRUE)   #> -0.10
skew(first_sem$caffeine_mg, na.rm = TRUE)   #>  1.74
skew(students$age, na.rm = TRUE)            #>  1.34

0 is symmetrical, positive a tail to the right. Between −1 and +1 is mild.

Most graduate students are in their twenties, fewer are older.

The normal distribution

Symmetrical and bell-shaped, described completely by its mean and SD.

Many tests assume data, or averages of data, follow it approximately (Chapter 7).

Checking sleep against the rule

z <- (sleep - mean(sleep)) / sd(sleep)
c(within_1_sd = mean(abs(z) < 1),
  within_2_sd = mean(abs(z) < 2),
  within_3_sd = mean(abs(z) < 3))
#> 0.707  0.958  0.995

Very close to 68-95-99.7.

z-scores give each value’s distance from the mean in standard deviations.

Q-Q plots compare with the normal

Sleep sits on the line. Caffeine bends away at the top: its long right tail.

Outliers are information first

Impossible values were set to NA in Chapter 3. Outliers are possible, just unusual.

An unusual value may be:

• an error that cleaning missed;
• a participant who misunderstood a question;
• a real case the research should pay attention to.

Find them and look at them before anything else.

Two rules for flagging

The box plot rule for caffeine

q1 <- quantile(first_sem$caffeine_mg, 0.25, na.rm = TRUE)
q3 <- quantile(first_sem$caffeine_mg, 0.75, na.rm = TRUE)
upper_fence <- q3 + 1.5 * (q3 - q1)
sum(first_sem$caffeine_mg > upper_fence, na.rm = TRUE)
#> [1] 35

35 students take more than the upper fence of 443 mg.

For sleep, roughly normal, the z-score rule flags 3 students.

The most extreme cases

first_sem |>
  filter(sleep_hours <= 4.5, caffeine_mg >= 600) |>
  select(student_id, sleep_hours, caffeine_mg, study_hours, wellbeing)

Nine students sleep 3.5 to 4.3 hours, take 610 to 900 mg of caffeine, and study up to 67 hours a week.

Nothing impossible: a real and worrying group, exactly what the thesis is about.

Never delete a value just because it is unusual

1. Check whether it is an error (Chapter 3).
2. If it is real, keep it and report it.
3. Choose summaries that are not thrown off by it, such as the median.
4. Check whether conclusions change without it: a sensitivity analysis.

Removing inconvenient values for a cleaner result is research misconduct.

Count the gaps first

semesters |> summarise(across(everything(), ~ sum(is.na(.x))))
students  |> summarise(across(everything(), ~ sum(is.na(.x))))

About 1% of each semester measurement is missing.

38 students did not answer the question about money worries.

Why data is missing matters more than how much

Students most worried about money may be the least willing to say so.

The biggest gap: students who left

left_ids <- setdiff(students$student_id,
                    semesters$student_id[semesters$semester == 3])
first_sem |>
  mutate(left_study = student_id %in% left_ids) |>
  summarise(students = n(), wellbeing = mean(wellbeing),
            considering_dropout = mean(considering_dropout == "Yes"),
            .by = left_study)

What attrition does to the later semesters

The students still present in semesters 3 and 4 were, on average, doing better.

A simple average of the later semesters overestimates how well students were doing.

• report how many left and how they differed;
• mixed models (Chapter 10) use every observation each student gave;
• multiple imputation exists for large gaps.

The correlation coefficient

\(r\) measures how closely two numeric variables follow a straight line, from −1 to +1.

Rough guides, not rules. A curved relationship can have \(r\) near zero.

A correlation matrix

first_sem |>
  select(stress, burnout, support, wellbeing, gpa) |>
  cor(use = "pairwise.complete.obs") |> round(2)

pairwise.complete.obs uses every student with both values for each pair.

A scatter plot matrix

pairs() draws one small scatter plot for each pair of variables.

Correlation is not causation

first_sem |>
  select(caffeine_mg, sleep_hours, gpa) |>
  cor(use = "pairwise.complete.obs") |> round(2)

flowchart LR
    S["Less sleep"] --> C["More caffeine"]
    S --> G["Lower GPA"]
    C -. "correlation −0.18" .- G

Caffeine may only look harmful because heavy users sleep less: a confounder (Chapter 8).

Describing the sample

first_sem |>
  summarise(students      = n(),
            female_pct    = 100 * mean(gender == "Female"),
            phd_pct       = 100 * mean(programme == "PhD"),
            part_time_pct = 100 * mean(study_mode == "Part-time"),
            age_mean      = mean(age, na.rm = TRUE),
            age_sd        = sd(age, na.rm = TRUE)) |>
  round(1)

Who took part: the first table in the methods chapter.

Every summary needs its spread and its n

A mean of 6.4 hours says little until the reader knows:

• whether students differ by minutes or by hours;
• whether it comes from 20 students or 600.

With missing values, the n differs from variable to variable, so report it for each.

A description table for the main variables

describe_var <- function(x) {
  c(n = sum(!is.na(x)),
    mean = mean(x, na.rm = TRUE), sd = sd(x, na.rm = TRUE),
    median = median(x, na.rm = TRUE), iqr = IQR(x, na.rm = TRUE),
    missing = sum(is.na(x)))
}
sapply(first_sem[, c("sleep_hours", "study_hours", "caffeine_mg",
                     "gpa", "wellbeing")], describe_var) |> t() |> round(2)

sapply() applies the function to each column; t() makes each variable a row.

Writing it up

The sample consisted of 600 graduate students (52% female; 29% PhD students), aged 23 to 52 (M = 29.7, SD = 4.8). Students slept 6.5 hours per night on average (SD = 1.0), and 65% slept less than 7 hours.

Every number comes from the code.

An exploration checklist

1. How many values are missing, and why?
2. What is the typical value, and how spread out are the values?
3. What is the shape: symmetrical or skewed, one peak or two?
4. Are there unusual values, and are they errors or real?
5. How does it relate to the other variables?

In your field: earth sciences

Old Faithful: eruptions last about 2 or about 4.5 minutes. The mean, 3.5, describes almost none of them. Report the two groups separately.

Practical lab: the Chapter 6 playground

Work through the playground exercises in your browser, with hints and solutions.

Every exercise also runs in RStudio, from the downloadable chapter project.

Practical exercises 1–3: summaries and shape

1. Mean, median, SD, and IQR of study hours: which pair would you report?
2. A Q-Q plot of wellbeing: does it look normal?
3. Count unusually high study hours with the box plot rule, and look at those students.

Practical exercises 4–6: gaps and relationships

4. Compare the sleep of students who left with those who stayed.
5. Classify the support–satisfaction correlation, and explain why it is not causal.
6. Choose and justify a summary for every variable in students.

Try this yourself

Take three variables from your own data.

• report centre, spread, n, and missing for each;
• draw each one’s shape and name it;
• flag unusual values and decide whether they are errors;
• write two sentences describing your sample.

Troubleshooting guide (Part 1)

Troubleshooting guide (Part 2)

Completion checklist

• missing values are counted, and their reasons considered;
• each summary suits the level and shape of its variable;
• every centre is reported with its spread and n;
• shapes are checked with graphs, not only numbers;
• unusual values are checked, kept if real, and reported;
• attrition is described;
• correlations are described as associations.

Misconceptions to leave behind (Part 1)

Misconceptions to leave behind (Part 2)

The chapter in one sentence

Describe every variable by its centre, spread, shape, gaps, and unusual values before you test anything.

Next: Chapter 7

The next chapter moves from describing the sample to inference:

• sampling distributions and standard errors;
• confidence intervals and the bootstrap;
• the logic of a test, built by shuffling;
• t-tests, effect sizes, and chi-square tests.

Questions

Which variable in your own data is most likely to be skewed?

Who is most likely to be missing from your data, and why?