These four terms come in two related pairs: population and sample describe the group being studied, while parameter and statistic describe numerical summaries.

Population. The population of a study is the entire group the collected data is intended to describe. If the population has $N$ units, we may write it as

$$\mathcal{P}=\{u_1,u_2,\dots ,u_N\},|\mathcal{P}|=N.$$

The units $u_i$ may be people, objects, or events. A variable assigns a measurement or category to each unit. If a quantitative variable gives the values $x_1,x_2,\dots ,x_N$ for the population units, then these values, rather than the units themselves, enter numerical calculations.

Sample. A sample is a smaller subset of the population, ideally one that is representative of the whole population. If the sample contains $n$ units, we often write their observed values as

$$x_1,x_2,\dots ,x_n,$$

where $n$ is the sample size.

Parameter. A parameter is a fixed numerical characteristic of a population. Parameters are usually denoted with Greek letters, such as the population mean

$$\mu =\frac{1}{N}\sum _{i=1}^N{x}_i.$$

Statistic. A statistic is a value calculated from the data in a sample. Statistics are often used to estimate population parameters. For example, the sample mean

$$\overline{x}=\frac{1}{n}\sum _{i=1}^n{x}_i$$

may be used to estimate the population mean $\mu$.

Variables are first divided into categorical and quantitative variables. Quantitative variables can then be described as discrete or continuous.

Categorical variable. A categorical variable, also called a qualitative variable, describes a category or characteristic. Categorical variables may be nominal or ordinal and cannot generally be added, multiplied, or averaged meaningfully.

Quantitative variable. A quantitative variable, also called a numerical variable, represents a measurable quantity numerically. Which arithmetic operations are meaningful depends on the variable's level of measurement.

Discrete quantitative variable. A discrete quantitative variable can take only specific values, often because it is counted.

Continuous quantitative variable. A continuous quantitative variable can take values across an interval, often because it is measured.

The levels of measurement describe what kinds of comparisons or calculations are meaningful for a variable. Four common levels are nominal, ordinal, interval, and ratio.

Nominal variable. A nominal variable is a categorical variable whose categories have no natural order.

Ordinal variable. An ordinal variable is a categorical variable whose categories have a natural order, but the differences between neighboring categories are not necessarily equal.

For interval and ratio variables, equal differences are meaningful: a change of the same number of units represents the same amount of change anywhere on the scale. The key difference is whether division comparisons are meaningful.

Interval variable. An interval variable is quantitative. Differences are meaningful, so it makes sense to subtract two values. However, the zero point is chosen as part of the scale. If the zero point were moved, the differences between values would stay the same, but division comparisons would change. Therefore, interval variables are compared by differences, not by statements such as "twice as large."

Ratio variable. A ratio variable is quantitative. Its zero point is fixed by the quantity itself, so measurements are counted from that starting point. Differences are meaningful, and division comparisons are meaningful too. This means we can say both "how much more" and "how many times as large."

The difference between interval and ratio variables is therefore a question about what comparisons the scale supports. Interval variables support subtraction. Ratio variables support both subtraction and division.

Frequency. Frequency is the number of times a data value, category, or interval occurs.

Relative frequency. Relative frequency is the frequency divided by the sample size $n$. It describes the proportion of the data represented by that value, category, or interval.

If a category or bin has frequency $f$, then its relative frequency is

$$\text{relative frequency}=\frac{f}{n}.$$

When relative frequency is reported as a percentage, we multiply by $100$:

$$\text{relative frequency percentage}=\frac{f}{n}\cdot 100\%.$$

A bar chart displays one bar for each category. The height or length of each bar indicates the frequency or relative frequency of that category.

A histogram groups numerical data into adjacent intervals, called bins. Its bars can show frequency or relative frequency. To prevent a boundary value from entering two bins, intervals may be written as half-open intervals such as $[0,100)$ and $[100,200)$; the final interval can include its right endpoint.

A symmetric distribution has left and right sides that are roughly balanced around the center.

A right-skewed distribution, also called a positively skewed distribution, has a longer tail to the right. This means that a few relatively large values stretch the distribution toward larger numbers.

A left-skewed distribution, also called a negatively skewed distribution, has a longer tail to the left. This means that a few relatively small values stretch the distribution toward smaller numbers.

Skewness matters because the mean is pulled toward the longer tail, while the median is usually less affected by extreme values.

For data values $x_1,x_2,\dots ,x_n$, the sample mean is

$$\overline{x}=\frac{1}{n}\sum _{i=1}^n{x}_i=\frac{x_1+x_2+\cdots +x_n}{n}.$$

Here, $x_i$ is the $i$th data value, $n$ is the sample size, and $\overline{x}$ is read as "x bar."

An outlier is a value much higher or lower than the rest of the data. Outliers may represent unusual but valid behavior, or they may be data entry errors.

The median is the middle value of an ordered data set.

To find the median:

1. Order the data values from smallest to largest.
2. If there is an odd number of values, the middle value is the median.
3. If there is an even number of values, the median is the mean of the two middle values.

If the ordered data values are

$$x_{(1)}\le x_{(2)}\le \cdots \le x_{(n)},$$

then the median $m$ is

$$m=\{\begin{array}{ll}{x}_{\left(\frac{n+1}{2}\right)},& \text{if }n\text{ is odd},\\ \frac{x_{\left(\frac{n}{2}\right)}+x_{\left(\frac{n}{2}+1\right)}}{2},& \text{if }n\text{ is even}.\end{array}$$

The mode is the data value or category that occurs most frequently in a data set.

The value with the largest frequency is always a formal maximizer in a finite data set. In this book, if every value occurs equally often, we say that the data set has no reported mode because no value is more frequent than the others. Otherwise, a data set may have one mode (unimodal), two modes (bimodal), or more than two modes (multimodal).

For grouped data, the modal class is the bin or interval containing the largest number of values.

If $f(x)$ denotes the frequency of the value or category $x$, then a mode is any value $x$ satisfying

$$f(x)=\underset{a}{\max }f(a).$$

The range is the difference between the maximum value and the minimum value of a data set:

$$r=x_{\max }-x_{\min }.$$

For a sample containing $n\ge 2$ values, the sample variance is the sum of the squared deviations from the mean divided by $n-1$:

$$s^2=\frac{{\sum }_{i=1}^n(x_i-\overline{x}{)}^2}{n-1}.$$

The divisor $n-1$ is used because the deviations are calculated from the sample mean: once $n-1$ deviations are known, the final deviation is determined by the requirement that they sum to zero. This adjustment makes $s^2$ suitable for estimating population variance.

For a sample containing $n\ge 2$ values, the sample standard deviation is

$$s=\sqrt{\frac{{\sum }_{i=1}^n(x_i-\overline{x}{)}^2}{n-1}}.$$

It measures the typical distance of the data values from the mean.

The $k$th percentile is a value that places approximately $k\%$ of the ordered data at or below it. Percentiles describe relative position within a data set.

Quartiles are common percentiles that split a data set into four parts:

• $Q_1$ is the first quartile, or 25th percentile.
• $Q_2$ is the second quartile, or 50th percentile. This is the median.
• $Q_3$ is the third quartile, or 75th percentile.

In percentile notation,

$$Q_1=P_{25},Q_2=P_{50},Q_3=P_{75}.$$

To find quartiles:

1. Order the data from smallest to largest.
2. Find the median, $Q_2$.
3. If the data set has an odd number of values, exclude the overall median before forming the lower and upper halves. If it has an even number of values, divide it into two halves of equal size.
4. Find the median of the lower half of the data; this is $Q_1$.
5. Find the median of the upper half of the data; this is $Q_3$.

The five-number summary is:

$$\text{Minimum},Q_1,\text{Median},Q_3,\text{Maximum}.$$

The interquartile range (IQR) measures the spread of the middle 50% of the data:

$$IQR=Q_3-Q_1.$$

A box plot, or box-and-whisker plot, is a graphical representation of the five-number summary.

In a box plot:

• the box spans from $Q_1$ to $Q_3$,
• a line inside the box marks the median,
• whiskers extend to the most extreme values that are not treated as outliers, and
• separate symbols may be used to mark potential outliers.

If no data values are shown separately as outliers, the whiskers extend to the minimum and maximum.

A box plot shows the center of the data, the spread of the data, and possible skewness or outliers.

A common rule marks data values as potential outliers if they lie below the lower fence or above the upper fence:

$$\text{lower fence}=Q_1-1.5\cdot IQR,\text{upper fence}=Q_3+1.5\cdot IQR.$$