Data Types and Levels of Measurement

Population vs sample

• Population: everyone you care about.
• Sample: the subset you actually measure.

All Singapore students = population. 200 surveyed students = sample.

Parameter vs statistic

A statistic estimates a parameter.

	Describes	Symbols	Known?
Parameter	Population	μ, σ	Usually unknown, fixed
Statistic	Sample	x̄, s	Calculated, varies by sample

Describes

• A parameter describes the whole population. A statistic describes only the sample you measured.

Symbols

• μ (mu) is the population mean and σ (sigma) is the population standard deviation. Greek letters usually mean “population”.
• x̄ (x̄ bar) is the sample mean and s is the sample standard deviation. Latin letters usually mean “sample”.

Known?

• A parameter is fixed, because the population has one true value, but you usually can’t know it. You’d have to measure everyone.
• A statistic is something you can calculate, but it changes depending on which sample you happen to draw.

Example: you want the average height of all Singaporean adults.

• Parameter: the true average across every adult is μ. It’s one fixed number, but you can’t realistically measure everyone.
• Statistic: you measure 200 random adults and get x̄ = 168 cm. Another random 200 might give 169 cm. The statistic changes from sample to sample, but each one is your estimate of μ.

That is what “a statistic estimates a parameter” means. You calculate the number you can (x̄) to estimate the number you can’t (μ).

The average height of all Singaporeans is μ (unknown). The average of your 200 respondents is x̄, which estimates μ.

Types of data

Qualitative = categorical. Quantitative = numerical.

Qualitative (categorical)

• Nominal: no order. Blood type, country, brand.
  • Binary is the 2-category case. Yes/no, pass/fail, zip code number
• Ordinal: ordered, but gaps are not measurable. Poor/fair/good, education level, 1st/2nd/3rd.

Quantitative (numerical)

• Discrete: countable whole numbers. Number of children, defects.
• Continuous: any value in a range, including decimals. Height, time, temperature.

The 4 levels of measurement (NOIR)

Level	Order	Equal gaps	True zero	Type	Example	What you can do
Nominal	No	N/A	No	Qualitative	Eye color	Count, mode
Ordinal	Yes	No	No	Qualitative	Satisfaction rating	Rank, median
Interval	Yes	Yes	No	Quantitative	°C, calendar year, IQ	Add, subtract, mean
Ratio	Yes	Yes	Yes	Quantitative	Weight, income, age	All of the above, plus ratios

Interval vs ratio Is “twice as much” meaningful? 20 kg is twice 10 kg (ratio). 20°C is not twice as hot as 10°C (interval).

Definition

Order – Can the values be ranked from low to high?

Order?	Example	Why
Yes	Poor < Fair < Good	Good is clearly “more” than poor
Yes	10°C < 20°C	Numbers sort naturally
No	Eye color, country	Blue is not “higher” than brown

Equal gaps – Is the step between neighbouring values the same size everywhere? If yes, subtraction is meaningful.

Equal gaps?	Example	Why
Yes	10°C → 20°C → 30°C	Each step is exactly 10°C
No	Race finish: 1st, 2nd, 3rd	1st may win by 0.1 s, 2nd beats 3rd by 5 s
No	Poor, fair, good	Is poor→fair the same jump as fair→good? Can’t say
N/A	Eye color	No order, so no gaps to compare

Quick test Pick three adjacent values A, B, C. Is the gap A→B the same as B→C? If you can’t say, the gaps are not equal.

Equal gaps only makes sense if there is order first.

True zero – Does zero mean “none of it”? If yes, ratios are meaningful.

• Weight: 0 kg = no weight, so 20 kg is twice 10 kg.
• Temperature in °C: 0°C is not “no heat”, so 20°C is not twice as hot as 10°C.

Common pitfalls

Likert scales and star ratings

• A 1-to-5 rating is ranked (5 > 4), so it’s ordinal. But nothing proves the jump from 1 to 2 feels the same as the jump from 4 to 5, so the gaps aren’t equal.
• Many analysts average ratings anyway, which treats them as interval. It’s common practice, but technically a stretch. Example: reporting “average rating 4.2” is convenient, but the median or the % of 4-5 ratings is safer.

Temperature

• °C and °F are interval. The gaps are equal, but zero is arbitrary (0°C is just the freezing point of water, not “no heat”). So 20°C is not twice as hot as 10°C.
• Kelvin is ratio. 0 K is absolute zero, where there’s truly no thermal energy. So 200 K really is twice as much thermal energy as 100 K.

Two separate classifications

• Discrete/continuous describes how the values arise: counted vs. measured.
• Interval/ratio describes the measurement scale: whether there’s a true zero.
• Every numerical variable gets one label from each pair.

Variable	Discrete or continuous	Interval or ratio
Number of children	Discrete	Ratio (0 = none)
Height	Continuous	Ratio
Temperature in °C	Continuous	Interval
Calendar year	Discrete	Interval