Summary statistics for univariate data

STAT1003 – Statistical Techniques

Dr. Emi Tanaka

Australian National University

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Table of contents

• R package datasets tips data
• Initial data analysis
• Statistical summary for univariate data
• Categorical variables
• Numerical summary for a categorical variable
• Graphical summary for a categorical variable
• Nominal vs. Ordinal variables
• Numerical variables
• Graphical summary for numerical variable
• Histogram
• A measure of central tendency
• Sample mean and median
• Skewness
• Modality
• Quantiles
• A measure of dispersion
• Measure of dispersions
• Boxplots
• Case study STAT1003 mark distribution
• Summary

R package datasets tips data

• R comes with many built-in datasets that are helpful for learning and practicing data analysis.
• Use the data() function to see available datasets and to load them.

The tips data from the GGally package is the tip a waiter received in one restaurant.

Initial data analysis

• When given a dataset, start with exploring the data.
• The tidyverse package is useful for this purpose (we will discuss this more later).

• What is the sample size?
• What is the observational unit?
• Which of the variables are categorical data? Which ones are numerical data?
• Classify the categorical variables as ordinal or nominal.

Statistical summary for univariate data

A statistical summary (or descriptive statistics) provides key numerical and graphical measures that concisely describe the main characteristics of a dataset.

Measures of Central Tendency

• Mean (average)
• Median
• Mode

Measures of Dispersion (Spread)

• Range
• Variance
• Standard deviation
• Interquartile range (IQR)

Tabular Summaries

• Frequency tables
• Contingency tables (cross-tabulations)

Graphical Summaries

• Histograms
• Boxplots
• Bar charts
• Scatterplots
• Etc

Categorical variables

There are two types of categorical data (or variable), referred to also as qualitative data:

• Nominal
  • no ordering or relationship
  • e.g. marital status, eye color, job, degree, race
• Ordinal
  • have a distinct ordering
  • e.g.,
    • ranking teacher as “poor/fair/good”,
    • survey answer “strongly disagree/disagree/agree/strongly agree”

Numerical variables can be transformed to or captured as ordinal variables, e.g.

• income brackets: [0, 1000), [1000, 2000), [2000, 3000), 3000+,
• age ranges: [0-18], (18-30], (30, 50), [50, 75), 75+.

Numerical summary for a categorical variable

Some useful numerical summary includes:

• Frequency (counts) of each category
• Relative frequency (proportion or percentage) of each category
• Mode: the most frequent occurring observation

Graphical summary for a categorical variable

• We use ggplot2 package for all data visualisation (taught in more detail later)

Bar charts / bar plots

Pie charts (avoid using)

Nominal vs. Ordinal variables

• We can use exactly the same statistics for ordinal data we used for nominal data, e.g., frequency tables, bar charts, pie charts, etc.

• For ordinal data, preserve the order of the categories.
• For nominal data, reorder the categories based on another variable (if appropriate).

Plot 1

ggplot(tips) +
  geom_bar(aes(y = day))

Plot 2

tips |> 
  mutate(day = reorder(day, day, length)) |> 
  ggplot() +
    geom_bar(aes(y = day))

Plot 3

day_order <- rev(c("Thur", "Fri", "Sat", "Sun"))
tips |> 
  mutate(day = factor(day, levels = day_order)) |> 
  ggplot() +
    geom_bar(aes(y = day))

How do these plots differ?

Numerical variables

There are two main types of numerical data:

• Continuous
  • measured in infinitely small increments
  • e.g. height, weight, portfolio returns, and stock prices
• Discrete
  • measured in fixed increments
  • e.g. number of cars you own, and number of heads in three coin flips

Some variables are continuous, but measured in a discrete manner, e.g. age (in years).

Graphical summary for numerical variable

For discrete data, we can use a barplot to visualise the distribution.

For continuous data, we can use a histogram to visualise the distribution.

Histogram

viewof binw = Inputs.range([0.01, 5], {step: 0.01, label: "bin width"})

The number of bins does affect the histogram appearance, so explore different values to see how it changes the plot.

A measure of central tendency

A measure of central tendency is a location of the “middle”, “center”, or “expected value” of the distribution of your data.

• Sample mean (or average) and median are examples of measures of central tendency

• What is the average customer tip?

Sample mean and median

The sample mean or average is:

\[\bar{x} = \frac{1}{n}(x_1+x_2 +\dots + x_n) = \frac{1}{n}\sum_{i=1}^nx_i.\]

The sample median is:

• middle number of the sorted observation when \(n\) is odd, and
• average of the two middle sorted observations when \(n\) is even.

Sample data: \[54, 71, 57, 70, 53\]

The (sample) mean is \[(54 + 71 + 57 + 70 + 53)/5 = 61.\]

Sorted sample data: \[53, 54, 57, 70, 71\]

So (sample) median is \(57\).

• The mean is commonly used
• But the median is more robust to extreme observations (outliers).

Skewness

• Skewness is a measure of asymmetry in a given distribution

Symmetric

Mean \(\approx\) Median

Positively skewed or
Right skewed

Mean > Median

Negatively skewed or
Left skewed

Mean < Median

Modality

The sample mode is the value with the highest frequency.

• Mode is useful for categorical data.
• For numerical data, mode is less useful as there may be no repeated values.
• However, we can look at the modality of a distribution: number of peaks in the distribution.

Unimodal distribution

Bimodal distribution

Multimodal distribution

Quantiles

A \(p\)-quantile is the value below which \(p\) (where \(0 < p <1\)) proportion of your data lie below.

• Note: quantiles do not need to be data values.
• Quartiles are special quantiles that divide the data into four equal parts:
  • First quartile (\(Q_1\)) or lower quartile is the 0.25 quantile
  • Second quartile (\(Q_2\)) or median is the 0.50 quantile
  • Third quartile (\(Q_3\)) or upper quartile is the 0.75 quantile

A measure of dispersion

A measure of dispersion/spread is a number representing the spread of data around a measure of central tendency.

• E.g. range, interquartile range (IQR), variance, standard deviation.

Measure of dispersions

• Sample deviation: the distance of an observation from its mean \(x_i-\bar{x}\)
• Sample variance: \[s^2 = \frac{1}{n-1}\sum_{i=1}^n (x_i - \bar{x})^2.\]
• Sample standard deviation: the square root of sample variance \(s\)
  • Conveys similar information as variance, but measure of units is the same as the data
• The range is the difference between the maximum and minimum values in the dataset.
• The interquartile range (IQR) is the difference between the third quartile and the first quartile (\(Q_3 - Q_1\)).

Population variance: \[\sigma^2 = \frac{1}{N}\sum_{i=1}^{N} (x_{i'} - \mu)^2.\]

Boxplots

L = \(Q_1 - 1.5 \times IQR\)
U = \(Q_3 + 1.5 \times IQR\)

• Boxplot do not work well for small datasets and certainly not for \(n < 5\).
• Boxplots are poor at showing multimodal distributions.

Case study STAT1003 mark distribution

How hard is STAT1003 at ANU for a typical undergraduate student?

Here is a sample assignment and quiz marks:

Five number summary: (55, 80, 88, 93, 100)

Mode: 6

• Note: five number summary is (minimum, \(Q_1\), median, \(Q_3\), maximum)
• What do you think based on the distribution of marks for assignment and quiz?

Summary

• Summary statistics describe main characteristics of the data

• Frequency table
• Mode
• Barplot

• Skewness
• Modality
• Quantiles
• A measure of central tendency: mean and median
• A measure of dispersion: range, IQR, variance and standard deviation
• Histogram
• Boxplot