Slides 📊

3.1. Introduction to Numerical Summaries: Notation and Terminology

Numerical summaries provide a concise way to convey a data set’s key characteristics. While visual summaries give a holistic view, numerical summaries offer precise, compact descriptions that are easy to compare, analyze, and report. We begin this chapter by introducing the language needed to discuss these numerical measures in depth.

Road Map 🧭

Establish the notation we’ll use throughout the course.
Distinguish between population parameters and sample statistics.

3.1.1. Notation: The Language of Statistics

Before diving into summary measures, we need a consistent language to describe our data. Statisticians use specific notation to represent variables, observations, and statistical measures.

Variables and Observations

We denote variables with lowercase letters. The most common choices are \(x, y,\) and \(z\), but we can use other letters if we have more than three variables or if others are more appropriate for the context.

Sample Size

The lowercase letter \(n\) indicates the number of observations in a sample.

Individual Data Points

We represent an observation from a variable with the variable name subscripted by an index. For example, if variable \(x\) has \(n\) observations, their values can be denoted with

\[x_1,\; x_2,\; x_3,\;\ldots,\; x_n\]

The subscripts prevent confusion when individual observations are discussed. The order of data collection doesn’t usually matter in assigning subcripts; what’s important is that each observation is referenced with a unique index in a consistent manner.

When we refer to a single arbitrary observation, we often use the letter \(i\) as the index. For example, to express the sum of all data points, we write \(\sum_{i=1}^n x_i\).

Example 1 💡: Putting the notations to use

Suppose a data set of weight measurements is collected, in pounds, from randomly selected patients in Hospital A. The dataset is:

\[\{301, 202, 101, 125, 131\}.\]

Use appropriate notation to describe the data set.

First, let \(w\) denote the variable of weight measurements. Since the data set contains five observations, \(n = 5\). Also,
- \(w_1 = 301\)
- \(w_2 = 202\)
- \(w_3 = 101\)
- \(w_4 = 125\)
- \(w_5 = 131\)
Write the formula of the arithmetic average (the sum of all values, divided by the number of values) in concise notation.

The formula can be written as

\[\frac{1}{n}\sum_{i=1}^n w_i\]

Multiple Groups

Sometimes we measure the same variable across different groups. In these cases, we use double subscript notation:

\[\bigl\{\,y_{11}, y_{12}, \ldots, y_{1n_1}\bigr\},\; \bigl\{\,y_{21}, y_{22}, \ldots, y_{2n_2}\bigr\},\; \ldots,\; \bigl\{\,y_{I1}, y_{I2}, \ldots, y_{In_I}\bigr\}\]

Here,

The first subscript (\(1, 2, \cdots, I\)) indicates the group.
The second subscript indicates the observation within a group.
Each group can have a different sample size, so we differentiate group sample sizes with subscripts: \(n_1, n_2, \cdots, n_I\).

Example 2 💡: Continue Practicing Notation

Suppose Hospital B and Hospital C also collected weight data from their randomly selected patients.

Hospital B: \(\{132, 215, 140, 149, 270, 192, 105\}\)
Hospital C: \(\{166, 128, 199\}\)

Use appropriate notation to represent the combined variable of weight measurements from Hospitals A, B, and C.

We still use \(w\) to denote the weight variable.
- Use group index of 1 for Hospital A. Then \(n_1 = 5\), and
  
  \[w_{11}=301, w_{12}=202, \cdots, w_{15}=131.\]
- Use group index 2 for Hospital B. \(n_2 = 7\), and
  
  \[w_{21}=132, w_{22}=215, \cdots, w_{27}=105.\]
- Use group index 3 for Hospital C. \(n_3 = 3\), and
  
  \[w_{31}=166, w_{32}=128, w_{33}=199.\]

We won’t use the double-subscript notation often in the early chapters, but it will become important when we study multi-sample methods in Chapters 11 and 12.

3.1.2. Parameter vs Statistic

Recall from Section 1.2:

A population is the complete set of individuals we are interested in studying.
A sample is a subset of the population that we observe and measure.

These definitions lead to another key distinction that will remain central throughout the course: population paramter vs sample statistics.

A sample statistic, or simply a statistic, is any quantity we compute from observed data to describe the sample.
Often, a statistic is also used as an estimate for a “true” quantity that charaterizes the population. This population-level quantity is called a parameter.

To clearly indicate which of the two we are referring to, we use different families of symbols. We write a population parameter with a Greek letter and a sample statistic with a Latin letter. See the table below for some important examples:

Population Parameter			Sample Statistic
Name	Notation	How to read	Name	Notation	How to read
Population Mean	\(\mu\)	“mu”	Sample Mean	\(\bar{x}\)	“x bar”
Population Median	\(\tilde{\mu}\)	“mu tilde”	Sample Median	\(\tilde{x}\)	“x tilde”
Population Variance	\(\sigma^{2}\)	“sigma squared”	Sample Variance	\(s^{2}\)	“s squared”
Population Standard Deviation	\(\sigma\)	“sigma”	Sample Standard Deviation	\(s\)	“s”

Avoid the common mistake ‼️

Since most metrics have both a population and a sample version, always be sure to specify which one you mean by using the full expression.

❌	✔
The mean is larger than the median.	The sample mean is larger than the sample median.
The variance is unknown, so we must use data to estimate it.	The population variance is unknown, so we must use data to estimate it.

3.1.3. Bringing It All Together

Key Takeaways 📝

Consistent notation is essential for clear communication in statistics.
Subscripts are used to identify specific observations by their indices.
Usually, Greek letters denote population parameters; Latin letters denote sample statistics.

3.1.4. Exercises

These exercises build your fluency with statistical notation and the critical distinction between population parameters and sample statistics.

Exercise 1: Basic Notation Practice

A quality control engineer measures the resistance (in ohms) of 6 randomly selected resistors from a production batch. The measurements are:

\[\{47.2, 46.8, 47.5, 47.1, 46.9, 47.3\}\]

What is the sample size? Use appropriate notation.
Let \(r\) denote the resistance variable. Write out the notation for each individual observation (\(r_1, r_2, \ldots\)).
Write the formula for the sum of all resistance measurements using summation notation.
Write the formula for the sample mean resistance using summation notation.
If the engineer wanted to denote the true average resistance of all resistors in the production batch, what symbol should be used? Why?

Exercise 2: Interpreting Summation Expressions

Given a dataset with \(n = 4\) observations: \(x_1 = 10, x_2 = 15, x_3 = 12, x_4 = 8\).

Evaluate each of the following expressions:

\(\displaystyle\sum_{i=1}^{4} x_i\)
\(\displaystyle\sum_{i=1}^{4} x_i^2\)
\(\displaystyle\left(\sum_{i=1}^{4} x_i\right)^2\)
\(\displaystyle\sum_{i=1}^{4} (x_i - 11)\)
Are your answers to parts (b) and (c) the same? Explain why this distinction matters.

Exercise 3: Parameter vs. Statistic Identification

For each description below, determine whether the quantity described is a population parameter or a sample statistic. Then provide the appropriate symbol.

The average GPA of all 45,000 students currently enrolled at Purdue University.
The average GPA calculated from a random sample of 200 Purdue students.
The standard deviation of tensile strength for 50 steel specimens tested in a lab.
The true proportion of all manufactured chips that are defective.
The median response time calculated from 1,000 recorded API calls to a server.
The variance of fuel efficiency across all vehicles of a particular model ever produced.

Exercise 4: Correcting Notation Mistakes

A student writes the following statements in their statistics report. Identify and correct any notation or terminology errors.

“The mean of our sample was μ = 75.3.”
“We calculated s² = 12.4 for the population variance.”
“The standard deviation is larger than expected.”
“Using summation notation: \(\sum_{i=1}^{10} = 250\)”
“The variance was σ² = 16, calculated from our 30 data points.”

Exercise 5: Multiple Groups Notation

A pharmaceutical company tests a new drug at three different dosage levels. The improvement scores (on a 0-100 scale) for patients in each group are:

Low dose (Group 1): 12, 15, 18, 14
Medium dose (Group 2): 25, 28, 22, 30, 27
High dose (Group 3): 35, 42, 38, 40, 45, 41

Write the sample sizes for each group using appropriate notation.
Using double-subscript notation, write the notation for the third observation in Group 2.
Write the formula for the mean improvement score in Group 1 using summation notation.
Write an expression for the total number of patients across all three groups.
Write the formula for the overall mean improvement score across all patients (hint: you’ll need to account for different group sizes).

Exercise 6: Conceptual Understanding

Answer the following conceptual questions about parameters and statistics.

Explain why we typically cannot know the exact value of a population parameter.
If we collect a sample and calculate \(\bar{x} = 50\), can we conclude that \(\mu = 50\)? Why or why not?
A researcher claims: “I surveyed every single employee in my company (all 150 of them) and found the average satisfaction score to be 7.2.” Should this be reported as \(\bar{x} = 7.2\) or \(\mu = 7.2\)? Explain.
Why do statisticians use different symbols (Greek vs. Latin letters) to distinguish parameters from statistics?
In the context of statistical inference, what is the relationship between sample statistics and population parameters?

Solution

Part (a): Why Parameters Are Typically Unknown

We typically cannot know the exact value of a population parameter because:

Populations are often very large or infinite — measuring every individual is impractical or impossible
Cost and time constraints — complete enumeration is usually too expensive
Destructive testing — some measurements destroy the item (e.g., testing battery life until failure)
Dynamic populations — populations may change while we’re measuring them

Instead, we use sample statistics to estimate population parameters.

Part (b): Can We Conclude μ = 50?

No, we cannot conclude that \(\mu = 50\) just because \(\bar{x} = 50\).

Reasons:

The sample mean \(\bar{x}\) is an estimate of \(\mu\), not the exact value
Different samples would likely produce different values of \(\bar{x}\)
Sampling variability means our estimate has uncertainty
We can only say that \(\bar{x} = 50\) is our best point estimate of \(\mu\)

Later chapters will introduce confidence intervals and hypothesis tests to quantify this uncertainty.

Part (c): Census vs. Sample

This should be reported as \(\mu = 7.2\) (population parameter).

Reason: The researcher surveyed every single employee in the company. This is a census, not a sample. When you measure the entire population of interest, the calculated value IS the population parameter, not an estimate of it.

Key consideration: The answer depends on how you define the population. If the population is “all current employees,” then 7.2 is \(\mu\). If the population is “all employees past, present, and future,” then it would be \(\bar{x}\).

Part (d): Why Different Symbols?

Statisticians use different symbols to:

Prevent confusion — clearly distinguish between known (calculated) and unknown (true) quantities
Communicate precision — readers immediately know whether a value is exact or estimated
Maintain mathematical rigor — formulas involving parameters vs. statistics often differ (e.g., dividing by \(n\) vs. \(n-1\))
Guide interpretation — helps researchers avoid overstating conclusions

Part (e): Relationship in Statistical Inference

In statistical inference:

Sample statistics are used to estimate population parameters
Statistics are observable and calculable from data
Parameters are typically unknown and fixed (though unobserved)
The goal of inference is to draw conclusions about parameters based on statistics
Methods like confidence intervals quantify how close we believe our statistic is to the parameter
Hypothesis tests assess whether data are consistent with hypothesized parameter values

The entire framework of statistical inference rests on this relationship: using what we can measure (statistics) to learn about what we want to know (parameters).

3.1.5. Additional Practice Problems

True/False Questions (1 point each)

The symbol \(\bar{x}\) represents the population mean.

Ⓣ or Ⓕ
If \(n = 5\) and the observations are \(x_1, x_2, x_3, x_4, x_5\), then \(\sum_{i=1}^{5} x_i\) represents the sum of all observations.

Ⓣ or Ⓕ
Greek letters are used to denote sample statistics.

Ⓣ or Ⓕ
The expression \(\sum_{i=1}^{n} x_i^2\) is equivalent to \(\left(\sum_{i=1}^{n} x_i\right)^2\).

Ⓣ or Ⓕ
A sample statistic is used to estimate a population parameter.

Ⓣ or Ⓕ
The notation \(s^2\) represents the population variance.

Ⓣ or Ⓕ

Multiple Choice Questions (2 points each)

A researcher calculates the average height of 100 randomly selected adults to be 68.5 inches. Which notation correctly represents this value?

Ⓐ \(\mu = 68.5\)

Ⓑ \(\bar{x} = 68.5\)

Ⓒ \(\sigma = 68.5\)

Ⓓ \(s = 68.5\)
In double-subscript notation, \(y_{34}\) represents:

Ⓐ The product of the 3rd and 4th observations

Ⓑ The 34th observation in the dataset

Ⓒ The 4th observation in the 3rd group

Ⓓ The 3rd observation in the 4th group
Which of the following correctly matches the symbol with its description?

Ⓐ \(\sigma\) — sample standard deviation

Ⓑ \(\tilde{x}\) — population median

Ⓒ \(\mu\) — population mean

Ⓓ \(s^2\) — population variance