Lecture 5
Describing Data

ABD 3e Chapter 3

Chris Merkord

Learning Objectives

• Differentiate between estimates of location and estimates of width
• Recognize that variability is not simply noise but is a key parameter that can be estimated
• Become familiar with the most common descriptive statistics
• Know when the mean or median is a more appropriate summary of location

Three Common Descriptions of Data

• Location (central tendency)

• Width (spread)

• Association* (correlation)

*later in term

We do statistics to learn about the World Out There (the population) from Data (the sample)

Estimates and Parameters

• We almost never sample an entire population. So we almost never know parameters from populations.
• But we can make educated guesses of parameters by making estimates from samples.

Histograms reveal location

Also called “center” or “central tendency” of values

Figure 1: Distribution of sepal length for the three Iris species, shown as histograms faceted by species. Source: iris dataset in R.

Histograms reveal location

Also called “center” or “central tendency” of values

Figure 2: Distribution of sepal length for the three Iris species, shown as histograms faceted by species. Red vertical lines indicate species-specific means. Source: iris dataset in R.

Choose the right measure of location

• Location summarizes where values tend to fall in a distribution.

• Different summaries answer different questions.

Mean

• Balance point of the distribution
• Strongly influenced by extreme values

Median

• Typical value for an individual
• Robust to skew and outliers

Mode

• Most common value or range
• Highlights peaks in the distribution

The mean

Estimate = \(\bar{Y}\)

“Sample mean”

Parameter = \(\mu\)

“Population mean”

\[ \bar{Y} = \frac{\sum Y_i}{n} \]

where

• \(\bar{Y}\) is the sample mean
• \(\sum\) denotes summation over all observations
• \(n\) is the number of observations
• \(Y_i\) is the observed value for the \(i^{\text{th}}\) individual

The median

The median is the middle value of an ordered dataset.

• Order the observations from smallest to largest
• If n is odd, the median is the \((n + 1)/2\)^th value
• If n is even, the median is the mean of the \(n/2\)^th and \((n/2 + 1)\)^th values

Source: Blythwood / Wikipedia

The mode

The mode is the most common value in a dataset.

• For numerical data, it usually represents a range of values (a peak in a histogram)
• A distribution can have one mode, multiple modes, or no clear mode
• Modes are useful for identifying common outcomes or clusters, not for summarizing overall center

Mean, median, and mode summarize location differently

• These measures summarize different aspects of location
• They coincide in symmetric distributions
• They diverge in skewed distributions
• Differences reveal the shape of the distribution, not noise.

Geometric visualisation of the mode, median and mean of an arbitrary probability density function. Credit: Cmglee, CC BY-SA 3.0, via Wikimedia Commons

There are multiple ways to measure width (variability)

Different measures of width emphasize different aspects of spread.

• Range
  Difference between the largest and smallest values

• Interquartile range (IQR)
  Spread of the middle 50% of values

• Variance and standard deviation
  Typical deviation from the mean

• Coefficient of variation (CV)
  Variability expressed relative to the mean

Range is the difference between the smallest and largest values

Use as a simple measure of spread, when working with small datasets, or as a quick exploratory metric

Example:

• Minimum value = \(1.25\)

• Maximum value = \(3.55\)

• Range = \(3.55−1.25=2.30\)

Because small samples tend to give lower estimates of the range than large samples, the sample range is a biased estimator of the true range of the population.

Interquartile range describes spread of the middle 50% of values

• Median = middle observation, partitions data into two halves

• Quartiles = Q1, Q2 (median), Q3, partition data into four quarters

• Interquartile range (IQR) =“Q” 3−“Q” 1

Example: \(Q3−Q1 = 3.045−2.34=0.705\)

Boxplots display the median, IQR, outliers

Example of a boxplot comparing groups

• IQR is the yellow span

• Location of the horizontal line (median) within the box indicates skew

  • Toward the top = left skew

  • Toward the bottom = right skew

• Points are outliers

  • Common definition of outliers: observations which lie more than 1.5× “IQR” from the edge of the box

Variance, standard deviation, and CV quantify spread around the mean

These measures quantify how far observations typically lie from the average.

• Observations fall both above and below the mean
• Simple differences cancel out, so variability needs a different approach
• Squared differences avoid cancellation and work well mathematically
• Variance and standard deviation measure absolute spread
• The coefficient of variation (CV) measures relative spread

Key idea:
Variability is a property of the data, not noise.

The variance

• Average squared distance of observations from the mean
• Emphasizes large deviations
• Expressed in squared units
• Numerator often called the “sum of squared differences” or the “sum of squares”.
• Parameter (true population value) \(\sigma^2\)
• Statistic (estimated from sample) \(s^2\)

Credit: Rod Pierce / mathisfun.com

\[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n} \]

\[ s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1} \] where

• \(x_i\) is the observed value for the \(i^{\text{th}}\) individual
• \(\mu\) is the sample mean
• \(\sum\) denotes summation over all observations
• \(n\) is the number of observations

Why \(n\) for population and \(n-1\) for sample? (Bessel’s correction)

• When you compute the sample mean, you’re letting the data pick the center that minimizes the squared distances.
• That makes the distances look smaller than they really are in the population
• So we slightly inflate the average squared distance by dividing by n−1 instead of n.
• Simply:
  • Population: we already know the true center, so we divide by \(n\)
  • Sample: we had to estimate the center first, so we divide by \(n - 1\)
• Most important for small samples
• What if we didn’t use correction? False impression of precision, harder to detect true differences between groups

The standard deviation

• Typical distance of observations from the mean
• Expressed in the same units as the data
• Larger values indicate greater spread
• Parameter (true population value) \(\sigma\)
• Statistics (estimated from sample) \(s\)

\[ \sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{n}} \]

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

The coefficient of variation

• Standard deviation relative to the mean
• Unitless measure of variability
• Useful for comparing spread across different scales

\[ \mathrm{CV} = \frac{s}{\bar{x}} \times 100\% \]

where

• \(s\) is the sample standard deviation
• \(\bar{x}\) is the sample mean
• \(\mathrm{CV}\) is expressed as a percentage

Choose the right measure of spread

Use different measures of variability depending on your goal.

Use variance when

• You are working with equations or statistical models
• Variability needs to be combined or compared mathematically

Use standard deviation when

• You want variability expressed in the same units as the data
• You are describing spread to a general audience

Use coefficient of variation (CV) when

• You are comparing variability across different units or scales
• Relative variability matters more than absolute differences

Rounding: when and how much?

Rounding affects how results are interpreted.

• Do not round intermediate steps. Keep full precision during calculations to avoid compounding error

• Round only final results. This preserves accuracy while improving readability

• Report appropriate precision. Typically one–two decimal places more than the original measurements

Goal: Communicate results clearly without implying false precision.

Example: Values are integers, report mean and SD w/ 1 decimal place

Binary data: the mean is a proportion

Binary outcomes can be coded as 0/1 (e.g., no/yes, absent/present).

• Code one outcome as 1 and the other as 0
• The mean equals the proportion of 1s
• This is equivalent to the fraction of observations with that outcome

Example: If 4 out of 5 individuals survive, the mean of the 0/1 data is 0.8 (80%).