Which scale of measurement has equal intervals and a true zero, so that ratios such as "twice as much" are meaningful?

Ratio scale, as for pollutant concentration

Interval scale, as for temperature in °C

Ratio scale, as for pollutant concentration

Ordinal scale, as for air quality index categories

Nominal scale, as for soil type

A distribution with a moment coefficient of kurtosis β₂ greater than 3 is described as

leptokurtic, sharply peaked with heavier tails than the normal

mesokurtic, exactly like the normal curve

leptokurtic, sharply peaked with heavier tails than the normal

Which statements about the Poisson distribution are correct? Select all that apply.

It describes the number of rare, independent events in a fixed interval

It is always symmetrical and bell-shaped

It describes the number of rare, independent events in a fixed interval

Assertion (A): The geometric mean is preferred to the arithmetic mean for summarising coliform counts. Reason (R): Such counts are typically right-skewed and approximately log-normal, so a few very high values distort the arithmetic mean. Choose the correct option.

Both A and R are true, and R is the correct explanation of A

Both A and R are true, but R is not the correct explanation of A

Which statements about Student's t-distribution are correct? Select all that apply.

It approaches the normal distribution as the degrees of freedom increase

Its variance equals 1 whatever the degrees of freedom

A set of soil-lead measurements has a mean of 40 mg/kg, a median of 36 mg/kg and a standard deviation of 12 mg/kg. Pearson's coefficient of skewness, 3(mean − median) ÷ s, is

−1.0, indicating negative skew

The syllabus text reads "t, 2 and F-distribution", a Greek letter having been lost. The distribution defined as the ratio of two independent variance estimates, each divided by its degrees of freedom, is

the χ² (chi-square) distribution

Statistical Approaches I: Attributes and Variables, Scales of Measurement, Central Tendency and Dispersion, Standard Error, Moments, Skewness and Kurtosis, Probability, Sampling Theory, and the Normal, Log-Normal, Binomial, Poisson, t, χ² and F Distributions

Unit IX gives the environmental scientist the statistical tools for reading monitoring data and building models. This chapter takes its first item: attributes and variables and their types; the four scales of measurement; measures of central tendency and dispersion, including the geometric mean for skewed environmental data; the standard error; moments, skewness and kurtosis; the basic concepts of probability; sampling theory; and the distributions the syllabus names, normal, log-normal, binomial, Poisson, t, χ² and F. The text of the official syllabus has lost a Greek letter in extraction, so its "2 and F-distribution" and "2-test" are to be read as the χ² (chi-square) and F distributions and the χ²-test. Research statistics are also taught in other courses on this platform; here the examples are environmental, and a standard error, a binomial probability and a skewness coefficient are worked.

1. Attributes and variables, and scales of measurement

An attribute is a qualitative characteristic that can only be classified or counted, such as the species of a tree, the land-use class of a pixel or whether a sample exceeds a standard. A variable is a characteristic that can be measured and varies from one observation to another. Quantitative variables are discrete, taking separate values, usually counts (the number of eggs in a nest, colonies on a plate), or continuous, taking any value in a range (temperature, concentration, rainfall). In an experiment or model the variable manipulated or used to predict is the independent variable, and the one that responds is the dependent variable.

Scale (Stevens, 1946)	Properties	Environmental example	Permissible statistics
Nominal	Categories with no order	Soil type, species, land-use class	Frequencies, mode, χ² tests
Ordinal	Ordered categories, unequal or unknown intervals	Air quality index categories, IUCN threat categories, a pollution-tolerance ranking	Median, percentiles, rank correlation
Interval	Equal intervals but an arbitrary zero, so ratios are meaningless	Temperature in °C, pH is often treated so	Mean, standard deviation, correlation
Ratio	Equal intervals and a true zero, so ratios are meaningful	Concentration, rainfall, mass, temperature in kelvin	All statistics, including geometric mean and coefficient of variation

2. Central tendency, dispersion and the standard error

The arithmetic mean is the sum of values divided by their number; it uses all the data but is pulled by extreme values. The median is the middle value when data are ordered and is robust to outliers; the mode is the most frequent value. Environmental concentrations are often right-skewed, with many low values and a few very high ones, and for such data, especially counts of bacteria, the geometric mean, the nth root of the product of n values (equivalently the antilog of the mean of the logarithms), is the preferred average: for coliform counts of 10, 100 and 1,000 per 100 mL the geometric mean is 100, while the arithmetic mean, 370, is dominated by the single high count. The harmonic mean suits averages of rates.

Dispersion measures the spread of data. The range is the difference between the largest and smallest values; the mean deviation averages the absolute deviations from the mean; the variance averages the squared deviations, s² = Σ(x − x̄)² ÷ (n − 1) for a sample, the divisor n − 1 correcting the bias of estimating the population variance from a sample; the standard deviation s is the square root of the variance, in the original units; and the coefficient of variation, CV = (s ÷ x̄) × 100, expresses spread relative to the mean, allowing variability to be compared between variables with different units or magnitudes. The standard error of the mean, SE = s ÷ √n, measures how precisely the sample mean estimates the population mean, and it shrinks as the sample grows: to halve it the sample must be quadrupled.

Dissolved oxygen (mg/L) x	Deviation x − x̄	Squared deviation
4	−2	4
8	2	4
6	0	0
5	−1	1
7	1	1
Sum 30, mean 6	Sum 0	Sum 10

For these five readings, s² = 10 ÷ 4 = 2.5, s = √2.5 ≈ 1.58 mg/L, CV = 1.58 ÷ 6 × 100 ≈ 26% and SE = 1.58 ÷ √5 ≈ 0.71 mg/L. The deviations from the mean always sum to zero, which is why they are squared.

3. Moments, skewness and kurtosis

The rth central moment of a distribution is the mean of the rth power of the deviations from the mean, μr = Σ(x − x̄)^r ÷ n. The first central moment is always zero; the second is the variance; the third measures asymmetry; and the fourth measures the weight of the tails and the peakedness. Skewness is measured by the moment coefficient γ₁ = μ₃ ÷ σ³ (or β₁ = μ₃² ÷ μ₂³): zero for a symmetrical distribution, positive when the long tail is to the right (mean > median > mode), as with most pollutant concentrations and income, and negative when it is to the left. Karl Pearson's coefficient of skewness uses the mean, mode and median: Sk = (mean − mode) ÷ s, or, because in moderately skewed distributions mean − mode ≈ 3(mean − median), Sk ≈ 3(mean − median) ÷ s. With a mean of 40, a median of 36 and a standard deviation of 12, Sk = 3 × 4 ÷ 12 = 1.0, a clear positive skew.

Kurtosis is measured by β₂ = μ₄ ÷ σ⁴, which equals 3 for the normal distribution; excess kurtosis, β₂ − 3, is therefore zero for a normal curve. A distribution with β₂ > 3 is leptokurtic, sharply peaked with heavy tails, meaning extreme values are more frequent than the normal curve predicts, a warning for environmental risk; one with β₂ < 3 is platykurtic, flat-topped with light tails; and one with β₂ = 3 is mesokurtic.

⚠️ Positive skew means the tail is to the right

In a positively skewed distribution the bulk of the data are at low values and a few high values stretch the tail to the right, so the mean exceeds the median, which exceeds the mode. Reading "positive" as "most values are high" reverses it. And a kurtosis coefficient of 3 is normal, not "zero kurtosis"; it is excess kurtosis that is zero.

4. Basic probability theory and sampling theory

Probability measures the likelihood of an event on a scale from 0 (impossible) to 1 (certain). It can be defined classically, as favourable outcomes divided by equally likely possible outcomes; empirically, as the long-run relative frequency; or axiomatically (Kolmogorov). The addition rule gives P(A or B) = P(A) + P(B) − P(A and B), which reduces to P(A) + P(B) for mutually exclusive events; the multiplication rule gives P(A and B) = P(A) × P(B) for independent events and P(A) × P(B | A) in general, where P(B | A) is the conditional probability of B given A. Bayes' theorem reverses conditional probabilities and underlies the updating of risk estimates with new evidence. Environmental applications include the return period of a flood, T = 1 ÷ p, where p is the annual probability of exceedance: a "100-year flood" has a 1% chance of occurring in any year, and the chance of at least one in 30 years is 1 − 0.99³⁰ ≈ 26%.

Sampling theory deals with drawing conclusions about a population, the whole set of units of interest, from a sample. A statistic computed from a sample (such as x̄) estimates a parameter of the population (such as µ). In simple random sampling every unit has an equal chance of selection; stratified random sampling divides the population into homogeneous strata (for example soil types or land-use zones) and samples each, improving precision; systematic sampling takes every kth unit or places points on a grid, common in field surveys; and cluster sampling samples groups of units. Non-random methods such as convenience and judgement sampling are cheaper but may be biased. The distribution of a statistic over all possible samples is its sampling distribution; by the central limit theorem, the sampling distribution of the mean approaches a normal distribution as sample size grows, whatever the shape of the population, with standard deviation equal to the standard error. The difference between a statistic and the parameter is the sampling error, reduced by larger samples; errors of measurement and bias are non-sampling errors that larger samples do not remove.

5. Distributions: normal, log-normal, binomial, Poisson, t, χ² and F

Distribution	Form and parameters	Environmental use
Normal	Continuous, symmetrical and bell-shaped, with mean = median = mode; parameters µ and σ; about 68.3% of values lie within ±1σ, 95.4% within ±2σ and 99.7% within ±3σ	Measurement errors, many physical quantities, sample means by the central limit theorem
Log-normal	The logarithm of the variable is normal; the variable is positive and right-skewed; summarised by the geometric mean and geometric standard deviation	Pollutant concentrations, particle sizes, bacterial counts, contaminant levels in soil
Binomial	Discrete: the number of successes in n independent trials, each with probability p; mean np, variance np(1 − p)	Number of samples out of n exceeding a standard; presence or absence of a species in plots
Poisson	Discrete: the number of rare, independent events in a fixed interval; P(x) = e^(−λ) λ^x ÷ x!; mean = variance = λ	Counts of organisms per quadrat under random dispersion, radioactive decays per second, accidents per year
Student's t	Symmetrical about zero like the normal but with heavier tails; one parameter, degrees of freedom (n − 1 for one sample); approaches the normal as degrees of freedom increase; introduced by W. S. Gosset ("Student", 1908)	Inference about means from small samples when σ is unknown
Chi-square (χ²)	The sum of squares of k independent standard normal variables; takes only positive values, right-skewed, with mean equal to its degrees of freedom	Goodness of fit, tests of independence in contingency tables, confidence intervals for a variance
F	The ratio of two independent χ² variables each divided by its degrees of freedom, that is, a ratio of two variances; two degrees-of-freedom parameters; named after R. A. Fisher	Comparing variances and analysis of variance (ANOVA)

Two worked cases. If each of four water samples independently has a probability of 0.5 of exceeding a standard, the probability that exactly two exceed it is C(4, 2) × 0.5² × 0.5² = 6 × 0.0625 = 0.375. If a normally distributed concentration has a mean of 50 µg/m³ and a standard deviation of 10 µg/m³, a value above 70 lies 2 standard deviations above the mean (z = 2), and the probability of exceeding it is about 2.3%. For a Poisson count with a mean of 2 per quadrat, the probability of an empty quadrat is e⁻² ≈ 0.135.

Key takeaways

Attributes are classified and variables measured; the scales are nominal, ordinal, interval (°C, arbitrary zero) and ratio (concentration, true zero), each permitting its own statistics.
The geometric mean suits skewed data such as coliform counts; s² = Σ(x − x̄)² ÷ (n − 1), CV = s ÷ x̄ × 100 and SE = s ÷ √n, which halves only when n is quadrupled.
The third moment gives skewness (positive when the tail is to the right, mean > median > mode) and the fourth kurtosis (β₂ = 3 normal, above 3 leptokurtic); Pearson's Sk ≈ 3(mean − median) ÷ s.
Probabilities add for mutually exclusive events and multiply for independent ones; a flood's return period is 1 ÷ p; stratified and systematic sampling are common in the field, and the central limit theorem makes sample means approximately normal.
Normal data lie 68.3/95.4/99.7% within 1/2/3σ; concentrations are often log-normal; binomial counts successes (mean np), Poisson rare events (mean = variance); t serves small samples, χ² goodness of fit and independence, and F the ratio of variances; the syllabus's "2" is χ².

Practice questions (10)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

Which scale of measurement has equal intervals and a true zero, so that ratios such as "twice as much" are meaningful?
1. Interval scale, as for temperature in °C
2. Ratio scale, as for pollutant concentration
3. Ordinal scale, as for air quality index categories
4. Nominal scale, as for soil type
Show answer
Answer: B — Ratio scale, as for pollutant concentration
A concentration of 20 mg/L is genuinely twice 10 mg/L because zero means none, which makes it a ratio scale. Temperature in °C has equal intervals but an arbitrary zero, so 20 °C is not twice as hot as 10 °C; ordinal and nominal scales lack equal intervals altogether.
Five dissolved-oxygen readings of 4, 8, 6, 5 and 7 mg/L have a sample standard deviation of √2.5 mg/L. What is the standard error of the mean in mg/L, to two decimal places?
Numerical answer — type the value.
Show answer
Answer: 0.71
SE = s ÷ √n = √2.5 ÷ √5 = √0.5 = 0.707, which is 0.71 mg/L. Reporting the standard deviation itself (1.58) confuses the spread of the data with the precision of their mean.
A distribution with a moment coefficient of kurtosis β₂ greater than 3 is described as
1. mesokurtic, exactly like the normal curve
2. leptokurtic, sharply peaked with heavier tails than the normal
3. platykurtic, flat-topped with light tails
4. positively skewed
Show answer
Answer: B — leptokurtic, sharply peaked with heavier tails than the normal
For the normal curve β₂ = 3; above 3 the distribution is leptokurtic, meaning extreme values are more likely than the normal curve suggests. Below 3 it is platykurtic. Kurtosis concerns tails and peakedness, while skewness concerns asymmetry.
Which statements about the Poisson distribution are correct? Select all that apply.
1. It is always symmetrical and bell-shaped
2. Its mean equals its variance
3. It describes the number of rare, independent events in a fixed interval
4. It is a discrete distribution
Show answer
Answer: B — Its mean equals its variance; C — It describes the number of rare, independent events in a fixed interval; D — It is a discrete distribution
Poisson counts are discrete, describe rare independent events such as decays or organisms per quadrat under random dispersion, and have mean = variance = λ, which is the basis of the variance-to-mean test of dispersion. For small λ it is strongly right-skewed, becoming nearly normal only for large λ.
Each of four water samples independently has a probability of 0.5 of exceeding a discharge standard. What is the probability that exactly two of the four exceed it? Give the answer to three decimal places.
Numerical answer — type the value.
Show answer
Answer: 0.375
By the binomial distribution, P(2) = C(4, 2) p² (1 − p)² = 6 × 0.25 × 0.25 = 0.375. Forgetting the six ways of choosing which two samples exceed gives 0.0625, the probability of one particular arrangement only.
Assertion (A): The geometric mean is preferred to the arithmetic mean for summarising coliform counts. Reason (R): Such counts are typically right-skewed and approximately log-normal, so a few very high values distort the arithmetic mean. Choose the correct option.
1. Both A and R are true, and R is the correct explanation of A
2. Both A and R are true, but R is not the correct explanation of A
3. A is true, but R is false
4. A is false, but R is true
Show answer
Answer: A — Both A and R are true, and R is the correct explanation of A
Both are true and R explains A. Averaging the logarithms, which is what the geometric mean does, treats the data on the scale on which they are roughly normal, so counts of 10, 100 and 1,000 give 100 rather than the arithmetic 370 set by one high count.
A normally distributed pollutant concentration has a mean of 50 µg/m³ and a standard deviation of 10 µg/m³. The probability that a single measurement exceeds 70 µg/m³ is about
1. 2.3%
2. 0.3%
3. 5%
4. 16%
Show answer
Answer: A — 2.3%
70 is two standard deviations above the mean (z = 2), and about 95.4% of a normal distribution lies within ±2σ, leaving about 2.3% in each tail. 16% is the tail beyond one standard deviation and 0.3% is the total beyond ±3σ.
Which statements about Student's t-distribution are correct? Select all that apply.
1. It takes only positive values
2. It is symmetrical about zero
3. It approaches the normal distribution as the degrees of freedom increase
4. Its variance equals 1 whatever the degrees of freedom
Show answer
Answer: B — It is symmetrical about zero; C — It approaches the normal distribution as the degrees of freedom increase
The t-distribution is symmetrical like the normal but has heavier tails, and it converges on the standard normal as degrees of freedom grow. Its variance, df ÷ (df − 2), exceeds 1 for small samples, and it takes negative as well as positive values; the only-positive distributions here are χ² and F.
A set of soil-lead measurements has a mean of 40 mg/kg, a median of 36 mg/kg and a standard deviation of 12 mg/kg. Pearson's coefficient of skewness, 3(mean − median) ÷ s, is
1. 0.33, indicating negative skew
2. 0, indicating symmetry
3. −1.0, indicating negative skew
4. 1.0, indicating positive skew
Show answer
Answer: D — 1.0, indicating positive skew
Sk = 3 × (40 − 36) ÷ 12 = 12 ÷ 12 = 1.0. The mean above the median shows a tail of high values to the right, typical of contaminant data; omitting the factor of 3 gives 0.33, and any positive value, not a negative one, follows from mean > median.
The syllabus text reads "t, 2 and F-distribution", a Greek letter having been lost. The distribution defined as the ratio of two independent variance estimates, each divided by its degrees of freedom, is
1. the t-distribution
2. the F-distribution
3. the Poisson distribution
4. the χ² (chi-square) distribution
Show answer
Answer: B — the F-distribution
The F-distribution is a ratio of two scaled χ² variables, that is, of two variances, and underlies ANOVA. The missing "2" in the syllabus is χ², itself the sum of squared standard normal variables; t describes a mean over its standard error.