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Statistical Analysis in Hydrology
Once hydrological or meteorological data have been collected, they must be checked, analysed,
and stored systematically. Historical records, including older datasets, remain a valuable
resource, particularly those that capture rare events such as major floods or extreme droughts.
These rare events often play a pivotal role in infrastructure design and water resources planning.
Use of Statistics in Hydrology
Statistics, in its simplest form, refers to numerical descriptors that capture important
characteristics of a dataset. These include:
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Mean – average value
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Median – middle value when data is ordered
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Standard Deviation – a measure of spread or variability
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Skewness – a measure of asymmetry in the data
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Correlation Coefficients – indicators of relationship between variables
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Confidence Limits – estimates of uncertainty around sample statistics
These metrics allow hydrologists to concisely summarise complex datasets and make informed
decisions based on data characteristics.
Probability in Hydrology
The term statistics also refers to a broader mathematical field in which probabilities are assigned
to the outcomes of uncertain events. These probabilities may be established through different
approaches:
1) Priori Probability – derived from known principles (e.g., the chance of getting heads in a coin
toss is 0.5).
2) Empirical Probability – based on repeated observations or experiments (e.g., distribution of
concrete cylinder strengths from past tests).
3) Bayesian Probability – combines subjective assumptions with observed data, updating
beliefs as more information becomes available.
Frequency Analysis in Engineering Hydrology
Frequency analysis is a cornerstone of engineering hydrology, particularly for:
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Estimating design flows and design rainfall
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Evaluating return periods for extreme events
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Establishing risk-based design criteria for infrastructure
A common method involves selecting a sample of observed data (e.g., peak annual flood flows)
and fitting a theoretical probability distribution to it. This fitted distribution is then treated as a
representative population from which future values can be inferred.
From the fitted distribution equation, values corresponding to specific probabilities (e.g., 1% AEP
flood) can be computed and used for design and risk assessment.
Commonly Used Probability Distributions in Hydrology
Log-Pearson Type III Distribution - Standard distribution used in the U.S. and Australia for peak
flood analysis
Extreme Value (EV) Distributions - Popular in Europe and gaining popularity in Australia; Wellsuited for modelling rare or extreme events (e.g., Gumbel distribution)
Other Applications of Statistics in Hydrology
Statistical methods also support:
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Regional Flood Frequency Analysis (RFFA)- Estimating design floods at ungauged
locations using regional relationships
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Correlation Analysis - Determining relationships between rainfall and runoff, or between
stations
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Trend and Variability Analysis - Detecting changes in climatic or hydrologic regimes over
time
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BASIC STATISTICS
Probability
Probability is a numerical expression of the likelihood or chance of an event occurring, with
values ranging from 0.0 (impossible event) to 1.0 (certain event). Probabilities express the
relative frequency of an event compared to all possible outcomes.
Probabilities may be derived in two main ways:
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A priori – based on logical reasoning or known principles.
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Empirical – based on analysis of observed data.
Key Rule: The sum of the probabilities of all possible outcomes must equal 1.0.
For example, the probability of a coin landing heads or tails is 0.5 each, assuming a fair coin
and equal likelihood.
Populations and Samples
In statistics, the data being analysed typically consist of values of a variable, such as streamflow,
rainfall intensity, or volume. These values may come from either a population or a sample:
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A population includes all possible occurrences of a variable under specific conditions.
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A sample is a subset of the population and includes only some of the possible values.
Example:
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Measuring the height of 30 people in a room gives you the population of that room.
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If you use those 30 people to estimate the average height of all Australians, the group is
now considered a sample.
Populations may be:
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Finite – e.g. number of people in a city.
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Infinite – e.g. potential compressive strengths of concrete, or theoretical flood peaks.
Important Considerations in Hydrology
In hydrology, sample data are often used to estimate population characteristics, such as:
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Mean
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Standard deviation
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Skewness
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Return periods
However, this practice introduces a risk of bias because sample statistics may not perfectly
represent the population—especially when the sample size is small. Therefore, correction
factors or bias adjustments are often required.
Understanding whether you are working with a true population or a sample is critical, as different
statistical formulas and interpretations apply in each case.
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Histograms and Frequency Distributions
In hydrology, variables such as rainfall, streamflow, and inter-event durations often occur over
a range of values, with some values occurring more frequently than others. This distribution of
values can be visualised using histograms or frequency distributions.
Histograms
A histogram is a bar chart that displays the frequency of occurrence of values:
- For discrete variables (e.g., number of flood events in a year), the number of occurrences of
each integer value is plotted directly.
- For continuous variables (e.g., time between storm events), the range of values is divided into
class intervals, and the number of values in each interval is plotted.
These
plots
reveal
key
characteristics
of
the
data,
- Cluster around a central tendency (e.g. mean),
- Accumulate toward the lower or upper end of the range, or
such
as
whether
values:
- Appear uniformly distributed.
Figure 1 Example Histogram
Frequency Distributions
By dividing the frequency in each class interval by the total number of values, a relative
frequency distribution is obtained. As sample size increases and class intervals become
narrower, the frequency distribution becomes smoother.
Mathematical curves can be fitted to frequency data to derive theoretical probability
distributions, which represent potential parent populations from which samples may be drawn.
Figure 2 Frequency Distribution
Cumulative Frequency Distributions
Cumulative distributions are generated by summing the frequencies from left to right across the
histogram or frequency table.
Figure 3 Cumulative Frequency Distribution
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Statistical Parameters
Measures of Central Tendency
These give the central value about which values tend to occur. Common measures include:
- Arithmetic mean (x̄ for sample, μ for population):
x̄ = ΣXi / n
where Xi = each value, n = number of values.
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- Weighted mean: if all values are not equally probable, the weighted mean can be used.
x̄ = Σ(Wi * Xi) / ΣWi
where wi values being weights or scaling factors applied to particular values in the data set.
- Median: Middle value in ordered data (or average of two middle values for even-sized sets).
- Mode: Most commonly occurring value.
For a symmetrical, single-peaked distribution, mean, median and mode are equal. In skewed
distributions, they differ.
Figure 4
Effect of Skewness
Measures of Dispersion
These indicate how much the data is spread out:
- Standard deviation (sample: s, population: σ):
s = sqrt[ Σ(Xi - x̄)^2 / (n - 1) ]
- Variance: s²
- Coefficient of variation: cv = s / x̄
For normal distribution:
- 68.3% of values within ±1 standard deviation
- 95.5% within ±2 standard deviation
- 99.7% within ±3 standard deviation
These can be used to set confidence intervals. These can be used to set confidence intervals. For
other theoretical distributions tables or formulae can be used to determine intervals or limits, inside
or outside of which certain events can occur. These confidence limits are a valuable aid in decision
making
Measures of Asymmetry
- Skewness (γx):
γx = [Σ(Xi - x̄)^3 / n] / s³
Positive skew indicates long right tail, negative skew a long left tail.
Figure 5
Values of Skewness
Measure of Peakedness (Kurtosis)
Kurtosis measures how peaked a distribution is relative to the normal distribution.
kurtosis = [Σ(Xi - x̄)4 / n] / s4
- Normal distribution kurtosis = 3.0
- kurtosis > 3: leptokurtic (peaked), kurtosis < 3: platykurtic (flat)
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Theoretical Distributions
Why Use Theoretical Distributions?
Fitting a theoretical distribution to data is useful because it helps us make better predictions. Instead
of relying only on one sample of data—which might have random errors or unusual patterns—we use
a smooth, general curve that represents the overall trend. This makes it easier to estimate
probabilities and calculate confidence limits.
Theoretical distributions are described using mathematical formulas. These come in different types,
depending on whether the data is continuous or discrete:
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For continuous data, we use something called a probability density function (PDF).
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For discrete data, we use a probability mass function (PMF).
One of the most common continuous distributions is the normal distribution. Its formula looks like this:
where: μ is the mean (average value), σ is the standard deviation (spread of the data).
Many other continuous distributions also have a similar format:
where C1 is a constant, and F(x) is a function of x.
Sometimes it’s hard to calculate probabilities from these functions because they involve complicated
integrals. To solve this, we often use pre-calculated tables or numerical methods (like Legendre
polynomials) on a computer.
Discrete distributions are often simpler. For example, the Poisson distribution, which is used for
counting events, has this formula:
where: x is a whole number (0, 1, 2, …); and λ is both the mean and the variance of the
distribution.
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Correlation and Regression
In statistics, we often need to understand how two or more variables are related. For example, we
might want to know how rainfall affects river flow, or how temperature changes with altitude.
Regression is a method used to find the best-fit relationship between variables. The most common
approach is called the least squares method. This method finds a line (or curve) that best fits the
data by minimizing the total squared difference between the actual values and the predicted values
of a dependent variable (usually called y). Figure 6.7 shows an example of a least squares fit.
Linear Regression
The simplest form of regression is a linear relationship between two variables:
y = a + b⋅x
where:
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y is the dependent variable (what you're predicting),
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x is the independent variable (what you're using to predict),
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a is the intercept (where the line crosses the y-axis),
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b is the slope of the line.
To find the values of a and b, we use the data and calculate:
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The sums of the variables: ∑x, ∑y,
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The sum of squares: ∑x2,
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The sum of the products: ∑xy.
These are plugged into formulas to calculate the best-fit.
Correlation Coefficient (r)
Once we fit a line, we want to know how strong the relationship is. This is measured by the correlation
coefficient, written as r.
It tells us how closely the data follows a straight line:
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r=+1: Perfect positive relationship
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r=−1: Perfect negative (inverse) relationship
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r=0: No relationship at all
Coefficient of Determination (r²)
Another useful statistic is the coefficient of determination, written as r2. It tells us how much of the
variation in y is explained by the regression equation.
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If r2=1, all the data points fall exactly on the regression line.
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If r2=0, the regression line explains none of the variation (data is completely scattered).
Non-Linear and Multiple Regression
Not all relationships are linear. Sometimes we can use transformations (like taking logarithms) to fit
non-linear relationships. For example:
logx = a + b⋅logy
This can be rewritten in original form as:
x=antilog(a)⋅yb
We can also use multiple regression to include more than one independent variable:
y = a + b⋅x1 + c⋅x2 + d⋅x3
This is useful when predicting something like river flow based on rainfall, temperature, and land slope
together.
Types of Correlation
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Cross-correlation looks at the relationship between two different variables.
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Serial correlation looks at how a single variable changes over time (like daily river flow).
Applications in Hydrology
In hydrology, correlation and regression are important tools. They are used to:
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Fill in missing data (e.g., missing rainfall records),
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Study relationships between variables like rainfall and runoff,
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Analyze time series, such as long-term streamflow records.
Understanding Confidence Limits
When we collect data and calculate things like a mean (average), we usually want to make
predictions or estimates about a larger group—not just the sample we measured. But because
samples can vary, our estimate might not be exact. That’s where confidence limits come in.
What Are Confidence Limits?
Confidence limits give us a range of values that we believe probably contains the true value (like the
true mean of the population). This range is based on the sample data, and we express how confident
we are in it using a confidence level—usually 95% or 99%.
For example:
If we say the mean rainfall is 100 mm ± 5 mm at 95% confidence, we are saying: We are 95%
confident that the true mean rainfall is between 95 mm and 105 mm.
That means if we repeated the experiment many times, 95% of the time the true mean would fall
within those limits.
Key Concepts
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Confidence Interval = Estimate ± Margin of Error
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Confidence Level = How sure we are (e.g., 95%)
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Wider intervals mean we are more cautious but less precise.
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Narrower intervals are more precise but might miss the true value more often.
What Affects the Width of Confidence Limits?
1. Sample size (n): A larger sample gives narrower (better) confidence limits.
2. Variability in the data (standard deviation): More variability = wider limits (less certainty).
3. Confidence level chosen:
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99% confidence = wider limits (more confidence, more room)
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90% confidence = narrower limits (less confidence, more precision)
Example: Confidence Limits for a Mean
Suppose you measured daily rainfall for 30 days and found:
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Mean = 80 mm
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Standard deviation = 10 mm
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Sample size = 30
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You want 95% confidence
Using a statistical formula (or calculator), you might find: Confidence Interval = 80 ± 3.6
So, the 95% confidence limits are 76.4 mm to 83.6 mm.
Why Are Confidence Limits Important?
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They show how reliable your results are.
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They help in making decisions under uncertainty.
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They are widely used in science, engineering, economics, and more.