Why does statistics exist?

You can’t weigh every ear in the field

You want to know the thousand-kernel weight of a wheat variety. Weighing all the millions of ears in the field is impossible. Instead you take a few hundred ears at random, weigh them, and generalize the result to the whole field.

All of statistics comes down to these two steps:

1. Understanding the data you have: organizing, summarizing and plotting the hundreds of numbers you measured. This is descriptive statistics, covered in Chapters 2 and 3.
2. Generalizing from your data to the whole: answering “The sample mean came out at 42 g, so what is the true mean of the field, and how sure am I?” This is inferential statistics. Hypothesis tests, confidence intervals and analysis of variance all belong here.

Population, sample and the numbers calculated from them

To avoid mixing up what each number stands for, you need this distinction:

	What does it cover?	What is a number calculated from it called?	Notation
Population	Every individual you are interested in (all the ears in the field)	Parameter	Mean $\mu$, standard deviation $\sigma$
Sample	The part of the population you actually select and measure (300 ears)	Statistic (estimator)	Mean $\bar{x}$, standard deviation $s$

The parameter is almost never known; what we have is always a sample. The $\bar{x}$ calculated from the sample is an estimate of the unknown $\mu$. Given as a single number it is a point estimate (“the mean is 42 g”); given as a range it is an interval estimate (“the mean is very probably between 40 and 44 g”).

Who is who in an experiment?

Imagine you set up a fertilizer trial: four nitrogen rates, each on three plots.

Term	In this trial
Treatment	What you are testing: the nitrogen rates
Independent variable	The variable you control: nitrogen rate
Dependent variable	The variable whose response you measure: yield (kg/da)
Experimental unit	The smallest unit a treatment is applied to: a single plot

This distinction decides which test you will choose later. “Does nitrogen rate affect yield?” asks about the effect of the independent variable on the dependent variable.

The type of data decides which calculations make sense

Not every number can be treated the same way. If you coded variety names as 1, 2, 3, saying “the average variety is 2.4” is meaningless. The scale on which you collected the data limits which statistics you can use:

Scale	What can you do?	Example	Usable measures
Nominal	Only “same / different”	Variety name, soil type, diseased yes/no	Frequency, mode
Ordinal	You can rank, but the gaps are not equal	Disease severity scale (0–5), quality grade	+ Median, quartiles
Interval	Differences are meaningful but zero is arbitrary	Temperature (°C)	+ Mean, standard deviation
Ratio	Zero really means “none”; multiples are meaningful	Yield, plant height, kernel count	+ Coefficient of variation, geometric mean

Note that 20 °C is not “twice as hot” as 10 °C: the zero point was chosen arbitrarily. With yield, however, 600 kg/da really is twice 300 kg/da.

There is also the distinction between discrete and continuous data. Data obtained by counting is discrete (kernels per ear: 31, 32, 33…). Data obtained by measuring or weighing is continuous (a plant can be 74.3 cm tall). This decides which chart you draw and which distribution you use.

Count or ratio?

Perhaps the most practical part of statistical literacy is noticing when a number you are shown is misleading.

Suppose four universities had these pass rates in an exam:

University	Sat the exam	Passed	Pass rate
A	60	45	75%
B	55	47	85%
C	62	46	74%
D	2	2	100%

University D looks like the winner at 100%, but only 2 people sat the exam. Had one more person sat and failed, the rate would drop to 67%. Rates calculated from small samples are very unstable. The reverse also holds: someone who looks only at the “number who passed” may think the school with the most students is the most successful. Don’t decide without seeing the raw count and the rate together.