Levels of Measurement and Frequency Tables

Main idea

Not all numbers mean the same thing: some are just labels, some only put things in order, and some can be subtracted or even divided meaningfully. Once you know what kind of data you have, a frequency table counts how often each value shows up, and dividing by the total turns those counts into proportions you can compare.

\text{relative frequency} = \frac{\htmlData{sym=0}{{f}}}{\htmlData{sym=1}{{n}}}

What the symbols mean

$f$
Frequency
How many times a value occurs.
$n$
Total
How many observations in all.

Explanation

A frequency table is simply a tally: each possible value, and how many times it showed up. The level of measurement is a label for what kind of arithmetic makes sense on your data, because a jersey number, a star rating and a weight are all "numbers" but mean very different things. The only formula, $\frac{f}{n}$ , is the count $f$ (how often a value appeared) divided by the total $n$ (how many observations in all), which turns a count into a fraction. That's genuinely all the math in this lesson.

Levels of measurement: each level allows everything the one above it does, plus a bit more.

Level	Meaning	Example
Nominal	Names only	Eye color
Ordinal	Ordered, gaps not equal	Star ratings
Interval	Differences meaningful, no true zero	Temperature in °C
Ratio	True zero, ratios meaningful	Weight, income

Frequency tables: list each value (or class), its frequency $f$ , its relative frequency $\frac fn$ , and its cumulative relative frequency, the running total of relative frequencies. Relative frequencies add to 1 (up to rounding), and the last cumulative value is 1.

Why bother with relative frequencies? Counts depend on how much data you collected. "12 defects" means one thing out of 50 and another out of 5,000. Dividing by $n$ puts tables of different sizes on the same scale.

A common mistake. It's tempting to say 20 °C is "twice as hot" as 10 °C. Temperature in Celsius is interval data: differences make sense, but 0 °C isn't "no heat," so ratios don't. Measure in kelvin, which has a true zero, and you'll see 293 K is nowhere near twice 283 K.

See it

50 server response times sorted into 50 ms bins. The bars are relative frequencies, tallest at 100–150 ms with 32% of requests, and the curve adds them up as you go. You can read percentiles straight off it: 70% of requests finish within 150 ms, and the curve passes 95% at around 244 ms. That's how engineers come up with claims like 'the 95th percentile is under 250 ms.'

Worked examples

Example 1

Twenty households report their number of pets: 0 pets (6), 1 pet (8), 2 pets (4), 3 pets (2). Build the relative and cumulative relative frequencies.

1
Divide each count by the 20 households: $\frac{6}{20}, \frac{8}{20}, \frac{4}{20}, \frac{2}{20}$ give $0.30, 0.40, 0.20, 0.10$ .

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

For the pets data, what is the relative frequency of exactly 1 pet?

Problem 2+5 XP

What is the cumulative relative frequency for 2 or fewer pets?

Problem 3+5 XP

What fraction of households have at least 2 pets?

Problem 4+5 XP

Temperature in °C is at which level of measurement?

Problem 5+5 XP

Movie ratings from 1 to 5 stars are at which level?

Problem 6+5 XP

A help desk logs 40 tickets: 18 software, 12 hardware, 6 network and 4 other. What is the relative frequency of network tickets?

Problem 7+5 XP

The response times in the picture fall in 50 ms bins with counts 5, 14, 16, 9, 4 and 2. What fraction of requests took less than 150 ms?

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

Estimate the median response time from the same table, assuming the times are spread evenly within each bin.

Where it's used

Survey dashboards, census reports and app analytics start with frequency tables; the level of measurement decides which averages and graphs are legitimate.

Reading a Latency HistogramStats · Computer Science

Sampling Methods and Bias Experimental Design and Ethics