Spread & Variation Lab

Percentiles

• Finding the kth percentile: sort the data and calculate the position with Position = (k ÷ 100) × (n + 1).
• Interpolation: if the position is a decimal (like 4.8), the value is between the 4th and 5th numbers. Take the lower number and add the decimal part times the difference between the upper and lower numbers. Example: 46 + 0.8 × (47 − 46) = 46.8.

On this tab, choose “L = (P/100)·(n + 1) — interpolate” to use that position formula.

Quick decision guide

Measure	Definition	Best used for	Outliers?
Mean	Sum ÷ number of values	Symmetric data without outliers	Highly sensitive
Median	Middle value of sorted data (50th percentile)	Skewed data, ordinal data, outliers (income, house prices)	Resistant
Mode	Most frequent value	Categorical (nominal) data, most common case	Not affected; may not be unique
Percentiles / quartiles	Values that cut sorted data into equal parts	Describing a position (“top 10%”) or spread	Resistant

• Percentiles split data into 100 parts, quartiles into 4 (Q1 = P25, Q2 = P50 = median, Q3 = P75), deciles into 10 (D1 = P10 … D9 = P90), quintiles into 5 (P20, P40, P60, P80).

Rules: summarizing data

• Frequency distributions: a table of classes (ranges) and how many values fall in each class.
• Cumulative frequency: add frequencies down the list to find the total up to a certain class.
• Relative frequency: the proportion of the total in one class. frequency ÷ total number of data points
• Cumulative relative frequency: the cumulative frequency up to a class, divided by the total.
• Midpoint: (lower limit + upper limit) ÷ 2. The population mean of grouped data is Σ(midpoint × frequency) ÷ N, shown under the table.
• Stem-and-leaf plots (stem = tens, leaf = ones) are on the Stem & Leaf tab. Left vs. right skew is on the Location tab.

How to read it

• Stem | Leaf. A row 1 | 2 5 8 means 12, 15, and 18.
• Median position: count every leaf to get n. Position = (n + 1) ÷ 2. If n is odd, that leaf is the median. If n is even, average the two middle leaves.

Levels of measurement

Classify each variable into one of these four levels.

Level	Rule	Examples
Nominal	Categories with no order	Eye color, types of cars
Ordinal	Categories with a meaningful order, but differences are meaningless	Letter grades A, B, C; rankings
Interval	Ordered, with meaningful differences, but no true zero	Temperature in Celsius, years on a calendar
Ratio	Ordered, with meaningful differences, and a true zero	Height, weight, income, age

Core vocabulary

• Sample (n): a subset of a larger group. Formulas use n − 1 in the denominator.
• Population (N): the entire group of interest. Formulas use N.
• Mean (x̄ or μ): the average. x̄ for a sample, μ for a population.
• Standard deviation (s or σ): typical distance of data from the mean, in the data's units.
• Variance (s² or σ²): the standard deviation squared, in squared units.

Measures of location

• Center: mean (Σx ÷ n), median (middle of sorted data, position (n + 1) ÷ 2), mode (most frequent; no mode / unimodal / bimodal / multimodal).
• Quantiles: percentiles (100 parts), quartiles (4), quintiles (5), deciles (10).
• Others: geometric mean (growth rates), harmonic mean (rates), midrange ((max + min) ÷ 2), trimean ((Q1 + 2Q2 + Q3) ÷ 4), winsorized mean (extremes replaced).
• Weighted mean = Σ(w × x) ÷ Σw. Moving average = average of a period and the k − 1 before it. Missing value = mean × n − known sum.
• Skew: right-skewed → Mode, Median, Mean (left to right); left-skewed → Mean, Median, Mode; symmetric → all equal.
• Which to use: mean for symmetric data without outliers; median for skewed data or outliers; mode for categorical data; percentiles to describe a position (“top 10%”).

Discrete vs. continuous data

	Discrete	Continuous
Values	Countable, separate values (often whole numbers)	Any value in a range, including fractions/decimals
Comes from	Counting	Measuring
Examples	Cars in a parking lot (no 2.5 cars), people in a room, a die roll (1–6)	Height, time, distance, temperature, weight

Why does recorded data always look discrete? Reality (time, distance, temperature) is often continuous, but data we write down is chopped into finite pieces:

1. Measurement resolution: rulers stop at millimetres, thermometers at 0.1°, so values get rounded (height 5.9 ft).
2. Computers are binary: everything is stored as a finite string of 1s and 0s.
3. Categorisation: we group continuous things into categories (“Under 18 / 18+”) so we can count and test them (e.g. Chi-squared tests).
4. Sampling: continuous signals are recorded at fixed intervals (a CD samples sound 44,100 times per second).
5. Storage: infinite detail can't be stored, so a photo becomes a grid of pixels.

So: a variable is truly discrete when it can only take countable values because you're counting (not just because a tool rounded it). Measured quantities stay continuous even if they're recorded to whole numbers.

Frequency tables

• Lower / upper class limit: the smallest / largest value that can belong to a class (e.g. 48 – 65).
• Class width = lower limit of next class − lower limit of this class (48 − 30 = 18).
• Class boundaries: split the gap between classes in half. Gap = next lower limit − this upper limit (102 − 101 = 1). Lower boundary = lower limit − gap ÷ 2 (101.5); upper boundary = upper limit + gap ÷ 2 (65.5).
• Midpoint = (lower limit + upper limit) ÷ 2.
• Cumulative frequency = this class's frequency plus all classes before it (12 + 3 + 4 = 19).
• Relative frequency = class frequency ÷ total frequency.
• Cumulative relative frequency = cumulative frequency up to that class ÷ total frequency.

Rounding rule (variance / standard deviation)

Round the final answer to one more decimal place than the original data. Data 4, 8, 12 → 1 decimal. Data 4.5, 8.2 → 2 decimals.

Which rule do I use?

Problem says…	Use
“bell-shaped” or “normal”	Empirical Rule (68–95–99.7)
Any / unknown shape	Chebyshev: 1 − 1/k²
“Which is more consistent?”	Coefficient of Variation — smaller wins

Study tips

• Get a good calculator. A statistical calculator (TI-30X or TI-84) saves time on variance, standard deviation, and correlation. Learn how to enter data into lists and run 1-Var Stats or LinReg.
• Watch your rounding. Instructions often say “round to two decimal places.” Do not round until the very last step. If the problem does not say, this app’s default is one more decimal place than the original data.
• Sample vs. population. If it says “sample,” use n − 1 for variance. If it says “population,” use N.

Test-taking checklist

1. Sample or population? Variance or standard deviation? (Variance is squared. Sample variance divides by n − 1; population variance divides by N.)
2. Bell-shaped (Empirical) or any shape (Chebyshev)?
3. Check rounding instructions — often two decimal places — and round only the final answer.
4. For consistency questions, compute CV for both groups; the smaller CV is more consistent.
5. Skew: left tail means Mean < Median < Mode; right tail means Mode < Median < Mean.
6. Levels: no order → nominal; order only → ordinal; differences but no true zero → interval; true zero → ratio.

Can't remember a formula? See the Mnemonics tab for memory phrases and their trigger words.