Summary Statistics for One Quantitative Variable

The concept in 60 seconds

A summary statistic turns a feature of recorded data into a number. For one quantitative variable, ask where the values are centered, how spread out they are, and where a value falls within the ordered data.

Use a graph alongside the summaries. Two very different distributions can share a mean and standard deviation. A summary table alone does not reveal every cluster, gap or detail of shape.

From a travel-time graph to numbers

Use the same fictional sample of $20$ Cedar High students from Topics 1.5 and 1.6. Each value is a student’s one-way home-to-school travel time on the Tuesday being studied, rounded to the nearest minute.

Ordered travel times, minutes:
$8$, $10$, $12$, $14$, $14$, $15$, $16$, $18$, $18$, $18$,
$20$, $22$, $24$, $25$, $26$, $28$, $30$, $32$, $35$, $42$

The summaries describe these $20$ sampled students on the specified Tuesday. They do not by themselves establish a population pattern or explain why a journey took longer.

Mean, median, quartiles and percentiles

Mean: share the total equally

The sample mean, written $\bar{x}$ (“x-bar”), is the total of the sample values divided by the sample size $n$.

For Cedar High, the total is $427 \,\mathrm{min}$, so $\bar{x}=\frac{427}{20}=21.35\,\mathrm{min}$. Imagine redistributing the total travel time equally among the $20$ observations: each would receive $21.35 \,\mathrm{min}$. The mean does not have to be an observed value.

Median: sort, then find the middle

1. Order the values from smallest to largest.
2. For odd $n$, take the value at position $\frac{n+1}{2}$.
3. For even $n$, average the values at positions $\frac{n}{2}$ and $\frac{n}{2}+1$.

With $20$ values, the middle positions are $10$ and $11$: $18$ and $20 \,\mathrm{min}$. The median is $\frac{18+20}{2}=19\,\mathrm{min}$. It separates the ordered data into a lower half and an upper half; ties may prevent exactly half from being strictly below it.

Quartiles: find the middle of each half

The median is also $Q_{2}$. The minimum, $Q_{1}$, median, $Q_{3}$ and maximum form a five-number summary: here $8$, $14.5$, $19$, $27$ and $42 \,\mathrm{min}$. Topic 1.8 turns this information into a boxplot.

Percentiles: locate a value within a group

The $p^{\mathrm{th}}$ percentile is a value at or below which approximately $p\%$ of the observations fall. The first quartile, median and third quartile correspond to approximately the $25^{\mathrm{th}}$, $50^{\mathrm{th}}$ and $75^{\mathrm{th}}$ percentiles.

When a question asks for the percentage at or below a specified value, count observations including ties at that value:

For Cedar High, $16$ of the $20$ travel times are at or below $28 \,\mathrm{min}$: $\frac{16}{20}\times100\%=80\%$. A travel time of $28 \,\mathrm{min}$ has an inclusive empirical percentile rank of $80$ in this sample.

If you are asked to calculate a percentile cutoff

Use the stated rank or interpolation rule. For example, if a question specifies the nearest-rank method, use position $\left\lceil\frac{pn}{100}\right\rceil$ in the ordered data, rounding the position upward. For $p = 80$ and $n = 20$, position $16$ gives $28 \,\mathrm{min}$. This optional percentile method is separate from the median-of-halves quartile method used here; do not switch conventions silently.

With small samples or tied values, a computed percentile cutoff need not place exactly $p\%$ of observations at or below it. Different accepted methods may give different cutoffs.

Range, $\operatorname{IQR}$, sample standard deviation and variance

Standard deviation: use distances from the mean

The sample standard deviation $s$ summarizes how spread out the observations are around $\bar{x}$. It is based on their squared deviations from the mean.

1. Calculate the mean $\bar{x}$.
2. For each observation, subtract the mean: $x_i-\bar{x}$.
3. Square each deviation, then add the squared deviations.
4. Divide by $n-1$ to get sample variance $s^{2}$.
5. Take the square root to get sample standard deviation $s$.

Consider $5$ fictional homework completion times: $12$, $14$, $16$, $18$ and $20 \,\mathrm{min}$. Their mean is $16 \,\mathrm{min}$.

The signed deviations add to $0$, so simply averaging them would hide the spread. Squaring prevents negative and positive deviations from cancelling. Taking the square root returns the standard deviation to the original units.

Units matter

• Mean, median, quartiles, range, $\operatorname{IQR}$ and standard deviation have the same units as the variable.
• Variance has squared units: $\,\mathrm{min}^{2}$, $\,\mathrm{dollar}^{2}$ or $\,\mathrm{point}^{2}$.
• Standard deviation is never negative. It is $0$ when all observations are equal.

For the Cedar High sample, $s^{2} \approx 78.66 \,\mathrm{min}^{2}$ and $s \approx 8.87 \,\mathrm{min}$. Keep extra digits during calculations and round the reported answer appropriately.

Why divide by $n-1$?

Once the sample mean is calculated, the deviations must add to $0$. If $n-1$ deviations are known, the last is determined. This is the idea behind the $n-1$ degrees of freedom used in sample variance.

If the task instead asks for the standard deviation of an entire population, the population formula uses its population size in the denominator. Do not choose the population output merely because a calculator shows it next to the sample output.

Two outlier rules—and why they can disagree

An outlier rule flags an observation that is unusually far from the others according to a specified criterion. Apply the rule in the question; report which rule you used.

For Cedar High, $42 \,\mathrm{min}$ is inside the $\operatorname{IQR}$ fences but above $\bar{x}+2s$. The two rules give different classifications. That is possible because one uses quartiles and the other uses the mean and standard deviation.

Resistance: how strongly can an extreme value change a summary?

A resistant statistic is less affected by extreme values. The median and $\operatorname{IQR}$ are resistant. The mean, range and standard deviation are not resistant; variance is also affected strongly by extreme values.

Resistant does not mean “never changes.” The median and $\operatorname{IQR}$ happen to stay the same in this example. Changing enough observations, or changing observations near the middle or quartile positions, can change them.

For a reasonably symmetric distribution without strong outliers, the mean and standard deviation are useful summaries. For a strongly skewed distribution or one with extreme outliers, the median and $\operatorname{IQR}$ usually give a more stable description of center and spread. Still report additional graph features when the question asks for them.

Four worked examples

Compare samples and explain what the numbers mean

A good response names the variable, group, statistic, numerical evidence and units. When comparing samples, use a direct comparison rather than two disconnected descriptions.

Compare independent samples

Two fictional offices record application processing times for independent samples of $9$ applications each. The observations are not paired. Using the stated quartile method:

This does not imply every B time is larger than every A time: B’s minimum is $5 \,\mathrm{min}$, below A’s $6$. The table alone also does not show the exact distribution shape or prove an office caused a delay. A claim about all applications would require information about the sampling process.

Justify your choice of statistics

Connect the choice to a feature of the distribution. For the alternative service times with an isolated $25$-minute value, write: “I would report the median of $7 \,\mathrm{min}$ and $\operatorname{IQR}$ of $4 \,\mathrm{min}$ because the distribution has a high extreme value, and these summaries are resistant to its effect.”

“The median is better” is incomplete. Explain the feature, the resistance, and the numbers. If two clusters are important, describe them as well; one center and spread pair can conceal the groups.

Calculator workflow: check the label and the data

1. Enter all observations into a list. On a TI-84 family calculator, use STAT → EDIT and enter the values in L1. Clear any old entries first.
2. Run 1-Var Stats from STAT → CALC, using L1. Leave the frequency list blank for a simple list containing each observation once.
3. Check $n$ and $\bar{x}$ first. For the $5$ homework times, $n$ should be $5$ and $\bar{x}$ should be $16$.
4. Read $S_x$ for sample standard deviation. For these times, $S_x \approx 3.1623$. The separate $\sigma_x$ output, about $2.8284$, uses the population denominator.
5. Square $S_x$ for sample variance if needed. Check $\mathtt{minX}$, $Q_1$, $\mathtt{Med}$, $Q_3$ and $\mathtt{maxX}$, remembering the quartile convention.

Menu wording varies by model. If using a frequency list, enter each distinct value with its count and check that $n$ equals the total count, rather than the number of distinct values. Keep calculator precision until the final reported answer.

Find and fix common mistakes

Eight practice questions with hints and solutions

Write a calculation and a sentence with units when the question asks for interpretation. All datasets below are invented teaching examples.

Quick revision checklist

Final understanding check

$8$ sampled visitors have recorded waits of $2$, $4$, $4$, $6$, $8$, $8$, $10$ and $22 \,\mathrm{min}$.

1. Calculate the mean, median, quartiles, range and $\operatorname{IQR}$ using our stated method.
2. The squared deviations from the mean total $272 \,\mathrm{min}^{2}$. Find $s^{2}$ and $s$.
3. Apply the $1.5\times\operatorname{IQR}$ rule. Choose a center/spread pair and justify your choice.
4. What percentage of waits is at or below $8 \,\mathrm{min}$? Convert the median and $\operatorname{IQR}$ to seconds.