Slides 📊

3.5. Choosing the Right Measure & Comparing Measures Across Data Sets

Now that we’ve explored different measures of central tendency and spread, we need to determine which measures are most appropriate for different types of data. The effectiveness of each measure depends on the distribution shape and the presence of outliers.

Road Map 🧭

Understand what it means for a measure to be resistant.
Examine how skewness affects different measures of center and spread.
Learn which measures to choose when the data is skewed or outliers are present.
Understand why non-resistant measures are often favored over the resistant ones.
Explore standardization for comparing observations across different datasets.
See how side-by-side boxplots can compare distributions across groups.

3.5.1. Resistant and Non-Resistant Measures

One of the most important considerations when choosing summary statistics is whether they are resistant to extreme values. This is because under their presence, certain measures lose their representative power for the data set.

A resistant measure is one that is not strongly affected by extreme values.
A non-resistant measure is a measure that is significantly influenced by them.

Skewed distributions provide an excellent way to understand the concept of resistance. Let’s examine what happens to our measures of center and spread when data is skewed.

Effect of Skewness on Numerical Summary Measures

Comparison of negatively and positively skewed distributions — Fig. 3.10 Skewed distributions

Observation 1: Sample mean is pulled towards the tail

In Fig. 3.10, notice how the sample means (solid dots) are pulled in the direction of the tail, while the medians remain firmly in the middle of the ordered data.

Observation 2: Many points in the tail are marked as explicit points

When data is skewed, box plots often flag many points as “explicit” in the direction of the tail. However, these flagged points don’t necessarily represent true outliers. They may simply be part of the natural tail behavior of the distribution.

Recall that real outliers typically show a clear gap from the rest of the data. These points should be investigated more thoroughly to determine if they are real outliers, but it is evident that many explicit points on Fig. 3.10 are not.

3.5.2. Choosing the Right Pair of Measures

Sample Mean vs. Sample Median

As we can see from the skewed distributions in Fig. 3.10,

The sample mean ($\bar{x}$) is non-resistant. This is because it gives equal weight to each observation, including the extreme values.
The sample median ($\tilde{x}$) is resistant because it depends only on the order (ranks) of most of the data, not the exact values.

Sample Variance (Standard Deviation) vs. IQR

Similarly, the measures of spread also differ in their resistance to outliers.

The sample variance ($s^2$) and sample standard deviation ($s$) are non-resistant because
- their computation depends on the sample mean, which is already non-resistant.
- their computation involves squaring the distances between data points and the sample mean, which amplifies the effect of extreme values.
The interquartile range (IQR) is resistant because it excludes the extreme values from computation by only considering the middle 50% of the data.

Why Use Non-Resistant Measures At All?

$\bar{x}$ and $s$ (or $s^2$) are still favored over $\tilde{x}$ and IQR when skewness and outliers are not a concern, because they carry important theoretical properties related to normality. These properties will form the foundation of many inference methods we develope later in the semester.

Summary

Property of data distribution	Resistance required?	Measure of center	Measure of variability
Skewed or has outliers	Yes	Sample median	IQR
Reasonably symmetric and has no outliers	No	Sample mean	Sample variance (sd)

3.5.3. Comparing Measures Across Data Sets

A. Side-by-Side Box Plots for Group Comparisons

When comparing a quantitative variable across categories, side-by-side box plots provide an excellent visualization tool.

Side-by-side box plots of heights by voice type — Fig. 3.11 Heights of singers in the New York Choral Society (1979) by voice type

This example shows the heights of singers in the New York Choral Society, categorized by voice type (soprano, alto, tenor, and bass). Some key observations are:

Singers with lower voice ranges tend to be taller.
There is a significant overlap in heights between tenors and basses, and between sopranos and altos.
Each voice type shows a different distribution shape of heights. For example, the distribution is negatively skewed for the soprano singers, while it is fairly symmetric for the altos except for a single outlier.

B. Standardization: Comparing Apples to Oranges

Often, we need to compare observations from different variables that use different scales or units. Standardization allows us to convert values to a common scale, making direct comparisons possible.

Suppose we have a data set $x_1, x_2, \cdots, x_n$, with a sample mean $\bar{x}$ and sample standard deviation $s$. Then for each $x_i$, its standardized value $z_i$ is calculated as:

\[z_i = \frac{x_i - \bar{x}}{s}.\]

The standardized values tell us how many standard deviations the data points are above or below the mean:

Positive $z_i$ indicates that $x_i$ is above the sample mean.
Negative $z_i$ indicates that $x_i$ is below the sample mean.
$z_i=0$ indicates that $x_i$ is equal to the sample mean.
$z_i = -2.25$ means the observation $x_i$ is 2.25 standard deviations below the sample mean.

Properties and Uses of Standardized Data

After standardization, the new data set $z_1, z_2, \cdots, z_n$ always has a sample mean of 0 and a sample standard deviation of 1 (this can be verified through a few algebraic steps). In other words, standardization adjusts any data set so that each value is measured relative to a common center and scale. This leads to several important properties:

Since they are unitless (the original units cancel out in the calculation), they provide an immediate sense of distance. For example, a data point with $z=-3$ is most likely very far from the central mass of the data.
They allow for direct comparison of observations from different distributions.

3.5.4. Bringing It All Together

Key Takeaways 📝

Resistant measures (median, IQR) are not strongly affected by outliers and skewness and should be used when data is skewed or contains outliers.
Non-resistant measures (mean, standard deviation) are influenced by extreme values but have better statistical properties for symmetric distributions without outliers.
The sample mean is always pulled in the direction of the tail in skewed distributions, while the sample median remains representative of the center.
Side-by-side box plots allow us to compare distributions of a quantitative variable across categories of a categorical variable.
Standardized observations enable comparisons across different distributions.

3.5.5. Exercises

These exercises build your skills in selecting appropriate summary measures based on distribution shape and comparing observations across different datasets using standardization.

Exercise 1: Identifying Resistant vs. Non-Resistant Measures

For each measure below, identify whether it is resistant or non-resistant to extreme values. Briefly explain why.

Sample mean ($\bar{x}$)
Sample median ($\tilde{x}$)
Sample standard deviation ($s$)
Interquartile range (IQR)
Sample range
Sample mode ($M$)

Exercise 2: Choosing Appropriate Measures

For each scenario, determine whether you should use (mean, standard deviation) or (median, IQR) as your summary measures. Justify your choice.

Heights of adult women in a large random sample (known to follow a normal distribution).
Home sale prices in a city where most homes cost between $200K–$400K but a few luxury homes sold for over $2 million.
Exam scores for a well-designed test where scores are roughly symmetric and range from 55 to 95.
Time spent on a website per visit, where most visits are 2-5 minutes but some users stay over an hour.
Battery life measurements for smartphones from a quality-controlled manufacturing process with tight tolerances.
Annual income for residents of a small town that includes both minimum-wage workers and a few multimillionaires.

Exercise 3: Effect of Skewness on Measures

A dataset of employee response times (in seconds) to IT help desk tickets has the following summary statistics:

Mean: 45.2 seconds
Median: 32.0 seconds
Standard Deviation: 38.5 seconds
IQR: 22.0 seconds

Based on the relationship between mean and median, what is the likely shape of this distribution?
Which direction is the distribution skewed? Explain how you know.
A manager wants to report “typical response time” to stakeholders. Which measure should they use? Why?
If the goal is to identify employees who are unusually slow to respond, would you use mean + 2SD or Q₃ + 1.5×IQR as a cutoff? Explain your reasoning.
If the five slowest response times (all over 150 seconds) were removed from the dataset, which measures would change more dramatically: (mean, SD) or (median, IQR)?

Exercise 4: Standardization Calculations

A student takes three different standardized tests:

Test	Student’s Score	Class Mean	Class SD
Statistics	78	72	8
Chemistry	85	80	10
Psychology	92	88	4

Calculate the z-score for each test.
On which test did the student perform best relative to their peers?
On which test did the student perform worst relative to their peers?
The student says “I scored highest in Psychology, so that’s my best subject.” Why might this conclusion be misleading?
If a z-score of 1.5 or higher earns an “A” in each class, in which classes would this student earn an “A”?

Exercise 5: Interpreting Z-Scores in Context

The fuel efficiency (mpg) of vehicles in a large fleet has a mean of 28 mpg and a standard deviation of 6 mpg.

A new hybrid vehicle gets 40 mpg. Calculate its z-score and interpret it.
An older truck gets 16 mpg. Calculate its z-score and interpret it.
The fleet manager wants to identify “exceptionally efficient” vehicles as those with z-scores above 2.0. What is the minimum mpg required?
The manager also wants to flag “inefficient” vehicles with z-scores below −1.5 for maintenance review. What is the maximum mpg that would be flagged?
A vehicle has a z-score of 0. What does this tell you about its fuel efficiency?
Is it possible for a vehicle in this fleet to have a z-score of −5? What would this mean, and is it likely?

Exercise 6: Comparing Across Groups with Box Plots

A company compares customer satisfaction scores (1-100 scale) across three store locations. The side-by-side box plots are shown below.

Box plots comparing satisfaction scores across three stores — Fig. 3.12 Customer satisfaction scores by store location

Based on the box plots:

Which store has the highest median satisfaction score?
Which store shows the most variability in satisfaction scores (based on IQR)?
Store B shows two explicit points on the low end. Should these customers be ignored in the analysis? Explain.
Store C appears to have a longer lower whisker than upper whisker. What does this suggest about the shape of Store C’s distribution?
A regional manager wants to identify the “most consistently satisfying” store. Which would you recommend based on both central tendency and spread?
If you could only examine one summary statistic for each store, would you prefer the mean or median for this comparison? Why?

3.5.6. Additional Practice Problems

True/False Questions (1 point each)

The sample mean is resistant to extreme values.

Ⓣ or Ⓕ
For a right-skewed distribution, the mean is typically greater than the median.

Ⓣ or Ⓕ
A z-score of −2.5 indicates the observation is 2.5 standard deviations above the mean.

Ⓣ or Ⓕ
The IQR is non-resistant because it depends on the sample mean.

Ⓣ or Ⓕ
After standardizing a dataset, the new mean is 0 and the new standard deviation is 1.

Ⓣ or Ⓕ
For symmetric distributions without outliers, the mean and median will be approximately equal.

Ⓣ or Ⓕ

Multiple Choice Questions (2 points each)

A dataset has Mean = 100 and Median = 120. This distribution is most likely:

Ⓐ Right-skewed (positively skewed)

Ⓑ Left-skewed (negatively skewed)

Ⓒ Symmetric

Ⓓ Bimodal
A student scores 75 on a test where the class mean is 70 and the standard deviation is 10. The student’s z-score is:

Ⓐ 0.5

Ⓑ 5.0

Ⓒ 7.5

Ⓓ −0.5
For heavily skewed data with several outliers, which pair of summary measures is most appropriate?

Ⓐ Mean and standard deviation

Ⓑ Mean and IQR

Ⓒ Median and standard deviation

Ⓓ Median and IQR
An observation has a z-score of 0. This means the observation:

Ⓐ Is an outlier

Ⓑ Equals the sample mean

Ⓒ Equals the sample median

Ⓓ Has zero variability