Week 10 – Continuous Variables

In earlier weeks we focused on categorical data and on how to describe it using:

• tables and bar plots

• measures of central tendency such as the mode and the median

• simple measures of spread such as the range and quartiles

This week, we slow down and revisit the same ideas, but now for continuous (numeric) data.
We will:

1. look at how to visualise continuous data with histograms

2. understand what the mean really is and how to calculate it

3. build up measures of spread: deviations, variance, and standard deviation

4. briefly describe the shape of a distribution (skew and kurtosis)

2. Central tendency

In the following examples, we are going to use data from 200 students on their IQ scores, study habits, sleep, stress, and related variables. These data are stored in the file Lecture1_data.csv, which we have already read in as students. Below is a quick numerical summary of some of the continuous variables:

students |>
select(IQ, Study_hours, Sleep_hours, Overall_mark) |>
summary()

       IQ         Study_hours     Sleep_hours     Overall_mark  
 Min.   : 85.0   Min.   :2.100   Min.   :4.200   Min.   :43.00  
 1st Qu.: 98.0   1st Qu.:4.500   1st Qu.:6.000   1st Qu.:57.00  
 Median :104.0   Median :5.900   Median :6.800   Median :68.00  
 Mean   :103.6   Mean   :5.845   Mean   :6.758   Mean   :66.24  
 3rd Qu.:109.2   3rd Qu.:7.200   3rd Qu.:7.600   3rd Qu.:74.25  
 Max.   :126.0   Max.   :9.900   Max.   :8.600   Max.   :86.00  

This summary shows basic information (minimum, quartiles, median, mean, and maximum) for some of the continuous variables we will use.

2.1 Revisiting mode and median

For central tendency we shall use:

• Mode: the most frequently occurring value.

• Median: the value for which 50% of observations are lower and 50% are higher (the middle value when data are ordered).

• Mean: the arithmetic average (sum of values divided by the number of values).

We previously applied these ideas to categorical variables, but they can also be used for numeric variables. The table below summarises which measures are appropriate for different types of data.

	Mode	Median	Mean
Nominal (unordered categorical)	✓	✗	✗
Ordinal (ordered categorical)	✓	✓	? (depends on context)
Numeric continuous	✓	✓	✓

For numeric variables there can be many different values, so the mode is not always very informative. However, we can still calculate it if we wish.

Below we look at the most frequent value (the mode) of the IQ variable in the students data.

students |>
count(IQ, name = "n") |>
arrange(desc(n))

# A tibble: 39 × 2
      IQ     n
   <dbl> <int>
 1   104    13
 2   108    12
 3   100    11
 4   101    11
 5    99     9
 6    95     8
 7   102     8
 8   106     8
 9   112     8
10   113     8
# ℹ 29 more rows

The first row of this output gives the most common IQ score in the sample.

Median for the IQ scores

Recall that the median is found by ordering the data from lowest to highest, and finding the mid-point.
In the students dataset we have IQ scores for 200 participants.
We find the median by ranking them from lowest to highest IQ and taking the mid-point between the two central scores.

We can also use the median() function:

median(students$IQ)

[1] 104

Mean

One of the most frequently used measures of central tendency for numeric data is the mean.

Mean:

The mean is calculated by summing all of the observations together and then dividing by the total number of observations (n).

When we have sampled some data, we denote the mean of our sample with the symbol\(\bar{x}\) (sometimes called “x-bar”).
The formula is:

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

Where:

• \(\bar{x}\) = estimate of the mean of the variable \({x}\)

• \(x_i\) = individual values of \({x}\)

• \(n\) = sample size

Samples and Populations

Statistics is about drawing conclusions about a population from a smaller set of sampled data.
A number we calculate from a sample is an estimate of the corresponding quantity in the population.

	Sample	Population
Number of observations	\(n\)	\(N\)
Mean	\(\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}\)	\(\mu = \frac{\sum_{i=1}^{N} x_i}{N}\)

In practice we almost always have a sample, so we work with \(\bar{x}\) and \(n\).

Calculating the mean for IQ

We can do the calculation by summing the IQ values and dividing by the number of students:

We can also check this directly in R:

• Option 1:

    sum(students$Overall_mark) / length(students$Overall_mark)

[1] 66.24

    mean(students$Overall_mark)

[1] 66.24

• Option 2

students |>
summarise(
mean_Overall_mark = mean(Overall_mark)
)

# A tibble: 1 × 1
  mean_Overall_mark
              <dbl>
1              66.2

All approaches give the same value; the last one is the style we will normally use when working with data frames.

Summarising variables

Functions such as mean(), median(), min() and max() can quickly summarise data.
We can combine them neatly using summarise() from dplyr.

summarise()

The summarise() function reduces variables down to one or more summary values.

# example structure

data |>
summarise(
summary_value1 = sum(variable1),
summary_value2 = mean(variable2)
)

So if we want to show the mean IQ and the mean Overall course mark of our students:

students |>
summarise(
mean_IQ          = mean(IQ),
mean_overall_mark = mean(Overall_mark)
)

# A tibble: 1 × 2
  mean_IQ mean_overall_mark
    <dbl>             <dbl>
1    104.              66.2

This produces a small table with one row and two columns: the average IQ score and the average overall mark in our sample.

3 Measuring spread: variance, and standard deviation

Knowing the mean is not enough. Two classes can have the same average mark but very different spread:

• in one class, everyone is close to the mean

• in the other, some students do extremely well and others very poorly

We therefore need a way to describe how much values vary around the mean.

For each observation we can look at how far it is from the mean. This difference is called a deviation:

Interquartile range

If we are using the median as our measure of central tendency and we want to discuss how spread out the spread are around it, then we will want to use quartiles (recall that these are linked: the quartile = the median).

We have already briefly introduced how for ordinal data, the 1st and 3rd quartiles give us information about how spread out the data are across the possible response categories. For numeric data, we can likewise find the 1st and 3rd quartiles in the same way - we rank-order all the data, and find the point at which 25% and 75% of the data falls below.

The difference between the 1st and 3rd quartiles is known as the interquartile range (IQR).
( Note, we couldn’t take the difference for ordinal data, because “difference” would not be quantifiable - the categories are ordered, but intervals are between categories are unknown)

In R, we can find the IQR as follows:

IQR(students$IQ)

[1] 11.25

# take the "students" dataframe |>
# summarise() it, such that there is a value called "median_IQ", which
# is the median() of the "IQ" variable, and a value called "iqr_age", which
# is the IQR() of the "age" variable.
students |> 
  summarise(
    median_IQ = median(IQ),
    iqr_IQ = IQR(IQ)
  )

# A tibble: 1 × 2
  median_IQ iqr_IQ
      <dbl>  <dbl>
1       104   11.2

Interpretation

• The median IQ in this sample is 104.
  This means that half of the students have an IQ below 104 and half have an IQ above 104.
  The median is a measure of central tendency that is not affected by extreme values.

• The interquartile range (IQR) for IQ is 11.25.
  This means that the middle 50% of students (those between the 25th and 75th percentiles) have IQ scores that differ by 11.25 points.
  The IQR is a measure of spread that tells us how tightly or loosely clustered the central half of the scores are.

In practical terms

• An IQR of 11.25 suggests that IQ scores in this class are moderately clustered around the centre.

• There is some variation, but most students fall within a relatively narrow band around the median.

Variance

If we are using the mean as our measure of central tendency, we can think about the spread of the data in terms of the deviations (the distance of each value from the mean).

Recall that the mean is denoted by \(\bar{x}\).
If we use \(x_i\) to denote the \(i^{\text{th}}\) value of \(x\), then the deviation for that value is:

\[ x_i - \bar{x} \]

The sum of deviations from the mean is always zero

The deviations from the mean always add up to zero:

\[ \sum_{i = 1}^{n} (x_i - \bar{x}) = 0 \]

The mean acts like a centre of gravity:
positive deviations (where \(x_i > \bar{x}\)) are exactly balanced by negative deviations (where \(x_i < \bar{x}\)).

Because deviations always sum to zero, they cannot be used directly as a measure of spread.
To solve this, we square them. Squaring makes all deviations positive, and larger deviations become disproportionately larger.
The average squared deviation is called the variance, written \(s^2\).

Variance: \(s^2\)

The variance is defined as the average of the squared deviations from the mean:

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

• \(s^2\) = sample variance
  
• \(\bar{x}\) = sample mean
  
• \(n\) = sample size

Why do we divide by \(n - 1\)?

The numerator

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

contains (n) deviations, but once the mean \(\bar{x}\) is calculated, the deviations are not independent.
They must sum to zero, meaning only (n - 1) of them contain unique information.

Example with two values

Suppose we only have two observations, \(x_1\) and \(x_2\).
The sum of squared deviations is:

\[ \sum_{i = 1}^{2} (x_i - \bar{x})^2 = (x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 \]

The mean for these two values is:

\[ \bar{x} = \frac{x_1 + x_2}{2} \]

Substituting this into the deviations:

\[ (x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 = \left( x_1 - \frac{x_1 + x_2}{2} \right)^2 + \left( x_2 - \frac{x_1 + x_2}{2} \right)^2 \]

This simplifies to:

\[ \left( \frac{x_1 - x_2}{\sqrt{2}} \right)^2 \]

Even though there are two points, there is effectively one independent piece of information, so we divide by \(n - 1 = 1\).
More generally, with \(n\) data points, we divide by \(n - 1\).

Calculating the variance in R

var() function:

students |>
  summarise(
    variance_overall_mark = var(Overall_mark),
    variance_IQ = var(IQ)
  )

# A tibble: 1 × 2
  variance_overall_mark variance_IQ
                  <dbl>       <dbl>
1                  97.0        77.2

Standard deviation

One difficulty in interpreting variance as a measure of spread is that it is expressed in units of squared deviations. It reflects the typical squared distance from a value to the mean. To bring the measure back into the same units as the original variable, we take the square root of the variance. This gives the standard deviation.

Standard Deviation: \(s\)

The standard deviation, denoted by \(s\), is an estimate of the typical distance of a value from the mean.
It is defined as the square root of the variance:

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

We can ask R to calculate the standard deviation of a variable using the sd() function:

students |>
summarise(
variance_IQ = var(IQ),
sd_IQ       = sd(IQ)
)

# A tibble: 1 × 2
  variance_IQ sd_IQ
        <dbl> <dbl>
1        77.2  8.79

4 Visualizations

Boxplots

A boxplot gives a quick picture of how the data are spread out.It provides a useful way of visualising the interquartile range (IQR). Here’s what each part of this plot shows:

Quartiles (Q1, Q2, Q3)

• Q1: 25% of the scores are below this value.

• Q2 (Median): the middle score in the dataset.

• Q3: 75% of the scores are below this value.

The dashed lines rise up from the box to show where each quartile sits.

The IQR (Interquartile Range)

• IQR = Q3 − Q1

• It shows the spread of the middle half of the scores.

• The pink bracket at the top highlights this range.

The Box

• The box stretches from Q1 to Q3.

• The line inside the box marks the median (Q2).
  This box tells you where most of the students’ IQ scores fall.

Whiskers (minimum and maximum values)

• The whiskers show the smallest and largest values in the data
  that are not considered outliers.

• The labels underneath show where these values are.

Possible outliers

• A value is considered an outlier if it is:

  • below Q1 − 1.5 × IQR, or

  • above Q3 + 1.5 × IQR.

The real dataset does not have outliers

Why this plot is helpful

A boxplot lets you quickly understand:

• the middle of the data (median),

• how spread out the scores are (IQR),

• whether the distribution is balanced or skewed,

• and whether any scores stand out as unusual.

Histograms

For categorical variables a bar plot works well because there are only a few distinct categories.
For a continuous variable such as Overall_mark, there may be many different values – sometimes every student has a slightly different mark.

If we tried to draw a bar for every distinct mark the plot would be messy and unreadable. Instead we group values into bins (ranges) and count how many observations fall within each bin. This gives a histogram.

ggplot(students, aes(x = Overall_mark)) +
geom_histogram(
bins  = 20,
colour = "white",
fill   = "steelblue4"
) +
labs(
x = "Overall course mark",
y = "Count",
title = "Histogram of Overall course mark"
)

Bins and their effect on shape

The bin width (or, equivalently, the number of bins) can change the apparent shape of the histogram:

• Few wide bins

# few wide bins

ggplot(students, aes(x = Overall_mark)) +
geom_histogram(
bins  = 5,
colour = "white",
fill   = "steelblue4"
) +
labs(x = "Height (cm)", y = "Count",
title = "Heights with 5 wide bins")

• Many narrow bins

# many narrow bins

ggplot(students, aes(x = Overall_mark)) +
geom_histogram(
bins  = 30,
colour = "white",
fill   = "steelblue4"
) +
labs(x = "Height (cm)", y = "Count",
title = "Heights with 30 narrow bins")

The data have not changed, but the visual impression can. When interpreting a histogram, always keep in mind that the choice of bins is under the analyst’s control.

Understanding Histograms Using Study_hours and Sleep_hours

We can use histograms to compare how spread out two continuous variables are. Below, we look at:

• Study_hours → wider spread

• Sleep_hours → narrower spread

Even before plotting, we can quantify their variability using the mean and standard deviation:

students |>
  summarise(
    mean_study = mean(Study_hours),
    sd_study   = sd(Study_hours),
    mean_sleep = mean(Sleep_hours),
    sd_sleep   = sd(Sleep_hours)
  )

# A tibble: 1 × 4
  mean_study sd_study mean_sleep sd_sleep
       <dbl>    <dbl>      <dbl>    <dbl>
1       5.84     1.79       6.76    0.983

The standard deviation tells us how much values typically vary around the mean.
Study hours vary more (SD ≈ 1.8), whereas sleep hours vary less (SD ≈ 1.0).

library(ggplot2)
library(patchwork)   # if not installed: install.packages("patchwork")

p1 <- ggplot(students, aes(x = Study_hours)) +
  geom_histogram(
    bins  = 20,
    colour = "white",
    fill   = "steelblue4"
  ) +
  labs(
    x = "Study hours per day",
    y = "Count",
    title = "Study_hours: Wider spread"
  ) +
  theme_minimal()

p2 <- ggplot(students, aes(x = Sleep_hours)) +
  geom_histogram(
    bins  = 20,
    colour = "white",
    fill   = "steelblue4"
  ) +
  labs(
    x = "Sleep hours per night",
    y = "Count",
    title = "Sleep_hours: Narrower spread"
  ) +
  theme_minimal()

p1 + p2   # displays them side-by-side

Interpretation

‘Study_hours’ and ‘Sleep_hours’ show very different spreads.
Even though the averages are not too far apart, Study_hours has a much larger standard deviation, meaning students differ widely in how long they study each day. This gives the histogram a wider base.

In contrast, Sleep_hours has a smaller standard deviation, meaning most students sleep roughly similar amounts. The histogram is therefore taller and narrower, because many values cluster close together.

Histograms are helpful because they let us see these differences in spread, not just calculate them.

Density Plot

A density plot (or density curve) shows the shape of a distribution using a smooth curve instead of bars. The y-axis represents density, and the total area under the curve equals 1.

p1 <- ggplot(students, aes(x = Study_hours)) +
  geom_density(fill = "steelblue4", alpha = 0.4) +
  labs(title = "Study_hours: Wider spread (SD ≈ 1.8)")

p2 <- ggplot(students, aes(x = Sleep_hours)) +
  geom_density(fill = "steelblue4", alpha = 0.4) +
  labs(title = "Sleep_hours: Narrower spread (SD ≈ 1.0)")

p1 + p2

Histogram	Density plot
Depends on bin width	No bins — smooth curve
Height = count	Height = density
Area is not 1	Area = 1
Can look quite different depending on bins	Much more stable visually

Note

When should you use each plot?

Use a histogram when you want to count how many observations fall in particular ranges. Use a density plot when you want to understand the overall shape of a distribution without worrying about bin choices. Using both together often gives the clearest picture.

5 Skewness

The purpose of this figure is to illustrate how the shape of a distribution affects the relationship between the mean, median, and mode. These three measures of central tendency behave differently depending on whether a distribution is positively skewed, symmetric, or negatively skewed.

1. What is skewness?

Skewness is a measure of asymmetry in a distribution:

• Positive skew (right-skewed)
  The distribution has a long tail on the right.
  A few very large values pull the mean upward.

• Symmetric distribution
  Data are balanced around the centre.
  Mean = Median = Mode.

• Negative skew (left-skewed)
  The distribution has a long tail on the left.
  A few very small values pull the mean downward.

Skewness is important because it affects which measure of central tendency best represents the “centre” of the data.

2. Why do the mean, median, and mode shift?

The mode always occurs at the highest peak—where most observations are concentrated.
The median is the midpoint value when the data are ordered.
The mean is sensitive to extreme values (outliers).

As a result:

• In positive skew:
  Mode < Median < Mean
  (the tail on the right pulls the mean to the right)

• In symmetric distributions:
  Mode = Median = Mean

• In negative skew:
  Mean < Median < Mode
  (the tail on the left pulls the mean to the left)

The ordering of these three values tells us about the direction of skew.

3. How to interpret the figure

This visual helps you identify skew by looking at:

• The direction of the tail

• Where the mean, median, and mode sit relative to each other

• The height of the density curve, which shows where values are most common

Being able to visually diagnose skew is essential in real data analysis because skew affects:

• Which summary statistics to use

• How reliable the mean is

• Which statistical models are valid

• How to correctly describe the data distribution

6 Choosing appropriate summaries

A quick guide for which measures to use:

Variable type	Centre (central tendency)	Spread (dispersion)
Categorical (nominal)	Mode	Frequency table
Categorical (ordered)	Mode / Median	Range / IQR
Continuous (numeric)	Mean / Median	Variance & Standard deviation (plus IQR)
Counts	Mode / Mean	Range, variance, standard deviation

7 Glossary

• Interquartile Range (IQR): The 3rd quartile minus the 1st quartile; the range of the middle 50% of the data.
• Mean: The sum of all observations divided by the total number of observations; the arithmetic average.
• Median: The middle value of an ordered dataset; 50% of observations lie below and 50% above.
• Mode: The most frequently occurring value in a dataset.
• Deviation: The distance from an observation to the mean value.
• Variance: The average squared distance of observations from the mean value.
• Standard deviation: The square root of the variance; the typical distance of observations from the mean.
• Quartiles (Q1, Q2, Q3): Values that split ordered data into four equal parts (25%, 50%, 75%).
• Interquartile fences: The cut-offs Q1 − 1.5 × IQR and Q3 + 1.5 × IQR used to flag potential outliers.
• Outlier: An observation that lies far from the bulk of the data, often beyond the 1.5 × IQR fences.
• Boxplot: Displays the median, quartiles, the IQR, and any potential outliers.
• Histogram: Shows the frequency of values that fall within bins of an equal width.
• Bin width: The size of the intervals used to group values in a histogram.
• Density curve / density plot: A smooth curve reflecting the distribution of a variable, where the total area under the curve is 1.
• Skew / Skewness: A measure of asymmetry in a distribution (positive/right skew or negative/left skew).
• Positive skew: A distribution with a long tail to the right; extreme high values pull the mean to the right of the median.
• Negative skew: A distribution with a long tail to the left; extreme low values pull the mean to the left of the median.
• Spread / dispersion: How much the values of a variable vary around a central value.
• Continuous variable: A numeric variable that can take many possible values on a scale (e.g., IQ, hours of sleep).
• Categorical variable: A variable that records group membership or labels (e.g., degree programme, gender).

• summarise() To summarise variables into one or more values according to whatever calculation we give it.
• mean() To calculate the mean of a numeric variable.
• median() To calculate the median of a numeric variable.
• IQR() To calculate the interquartile range for a numeric variable.
• sd() To calculate the standard deviation of a numeric variable.
• var() To calculate the variance of a numeric variable.
• min() / max() To obtain the smallest and largest values of a numeric variable.
• count() To count how many times each value or category occurs.
• ggplot() To start a plot using the ggplot2 grammar of graphics.
• geom_histogram() To add a histogram to a ggplot.
• geom_boxplot() To add a boxplot to a ggplot.
• geom_density() To add a density curve to a ggplot.
• geom_vline() To add a vertical reference line (e.g., for a mean or median) to a ggplot.

8 Key commands (continuous variables)

Use this table as a quick reference for the most common functions used in Week 10 when describing numeric variables.

Purpose	What you are doing	Example command(s)
Inspect data	Check variable types and names	glimpse(), str()
Select variables	Keep only the columns you need	select(IQ, Study_hours)
Quick numeric summary	Get min, quartiles, median, mean, max	summary(students$IQ)summary(select(students, IQ, Sleep_hours))
Count values (discrete numeric)	Find repeated values (e.g. mode candidates)	`count(IQ, name = “n”)
Mean	Calculate average	mean(students$IQ, na.rm = TRUE)
Median	Calculate midpoint	median(students$IQ, na.rm = TRUE)
Minimum / maximum	Find extremes	min(students$IQ, na.rm = TRUE)max(students$IQ, na.rm = TRUE)
Quartiles	Get Q1, Q2 (median), Q3	quantile(students$IQ, probs = c(.25, .5, .75), na.rm = TRUE)
IQR	Measure spread of middle 50%	IQR(students$IQ, na.rm = TRUE)
Variance	Measure spread around the mean (squared units)	var(students$IQ, na.rm = TRUE)
Standard deviation	Typical distance from mean (original units)	sd(students$IQ, na.rm = TRUE)
Summarise in a data frame	Produce clean summary tables	summarise(mean_IQ = mean(IQ), sd_IQ = sd(IQ))
Summarise by groups	Compare numeric summaries across categories	`group_by(Control)
Histogram	Visualise distribution (bins)	ggplot(students, aes(x = IQ)) + geom_histogram(bins = 20)
Boxplot	Visualise median, IQR, and outliers	ggplot(students, aes(x = "", y = IQ)) + geom_boxplot()
Density curve	Smooth view of distribution shape	ggplot(students, aes(x = IQ)) + geom_density()
Side-by-side plots	Combine plots for comparison	p1 + p2 (with patchwork)
Missing values	Check how many values are missing	sum(is.na(students$IQ))
Remove missing values	Keep only complete observations	filter(!is.na(IQ))