Chapter 21 Distributions Statistical Summaries and Geological Sampling

Year 1 · Mathematics & Statistics · G117-L3

Calculate central tendency and spread, read distributions, compare sampling designs and evaluate a worked geological dataset without overgeneralizing.

A repeatable workflow from a testable question to an evidence-based interpretation. Original Geology Hub teaching diagram.

Learning objectives: Define population and sample, calculate basic summaries, choose suitable distribution displays, and explain why sampling design limits geological inference.

Statistics begins before calculation

A population is the full set of objects or observations targeted by a question. A sample is the subset actually studied. A statistic summarizes a sample; a parameter describes a population. If the question concerns an entire formation, measurements from one convenient roadside exposure may not represent its full variation. A precise calculation cannot repair an unrepresentative sampling design.

Qualitative variables describe categories such as lithology, while quantitative variables describe counts or measurements. Discrete counts differ from continuous measurements. Keep units and measurement methods with the data. Grouping observations into classes can reveal patterns but also hide detail; results may change when class boundaries change. Preserve the original ungrouped values so alternative summaries remain possible.

Display the distribution, then summarize it

A frequency table counts observations in each category or interval. Relative frequency divides counts by the total, and cumulative frequency accumulates counts through ordered intervals. Histograms show quantitative distributions through adjacent bins; frequency polygons connect class-related values. A cumulative curve helps locate percentiles. Bar charts of categories and histograms of numerical intervals serve different purposes, even when their shapes look similar.

The arithmetic mean uses all values, the median identifies the middle of ordered data and the mode identifies the most frequent value or category. The geometric mean is useful for some positive multiplicative data, while the harmonic mean applies to particular rate-averaging situations. Do not select an unfamiliar mean simply because its result looks convenient. The quantity and sampling process determine which summary makes sense.

Frequency displays answer different questions

A frequency counts observations in an interval. Relative frequency divides by the total, while cumulative frequency adds all observations at or below an ordered threshold. For a fictional sample of 20 grains, six in one size class give a relative frequency of 0.30. A histogram displays class counts or an explicitly defined density; a frequency polygon connects class-related values. A cumulative curve helps locate thresholds such as the median. Changing bin widths without adjusting the interpretation can make visual comparisons misleading.

Different means suit different mathematical relationships

The arithmetic mean sums values and divides by their count. A geometric mean of positive values works through their product or mean logarithm: two illustrative sizes of 2 and 8 millimetres have geometric mean 4 millimetres. A harmonic mean can describe appropriately weighted rate problems. If equal thicknesses accumulate at 2 and 6 millimetres per year, their combined average rate is 3 millimetres per year, because the slower interval consumes more time. Averaging those rates arithmetically would answer a different weighting question.

Spread needs a definition and units

The range uses extremes and can be strongly influenced by one observation. Quartiles divide ordered data into portions; quartile deviation is half the interquartile range under the convention used. Mean absolute deviation averages absolute differences from a stated centre, whereas standard deviation uses squared deviations before taking a square root. Standard deviation has the original measurement units; variance has squared units. A coefficient of variation divides standard deviation by a suitable nonzero mean, commonly expressing the result as a percentage.

Sampling design determines the population represented

A simple random design uses a defined random selection mechanism over eligible units. A stratified design deliberately represents defined groups, while systematic sampling uses regular spacing after a suitable start. If bedding alternates periodically, a poorly chosen regular spacing can repeatedly sample one bed type and miss another. Nearby geological measurements may also be correlated. Ten adjacent points on one channel deposit do not necessarily provide the same regional information as ten appropriately distributed independent observations.

Connect numerical summaries to geological questions

Imagine two sandstone samples with identical mean grain size. One is tightly concentrated around that mean; the other mixes fine and coarse populations. Their sorting and process interpretations can differ despite the matching average. Examine distributions before assigning one environmental label. Report the sample size, measurement method and selection rule alongside centre and spread. These details establish whether the contrast concerns the sampled specimens, a mapped facies or a broader formation, preventing a correct calculation from becoming an unjustified regional conclusion.

Worked centre and spread calculation

Use fictional pebble diameters of 2, 3, 3, 4 and 8 centimetres. Their sum is 20, giving a mean of 4 centimetres. The median is 3 and the mode is 3. The range is 8 − 2 = 6 centimetres. The largest value pulls the mean above the median, revealing why reporting only one “average” conceals the distribution.

Deviations from the mean are −2, −1, −1, 0 and 4; squared deviations sum to 22. The sample variance is 22/(5 − 1) = 5.5 square centimetres, so sample standard deviation is about 2.35 centimetres. Dividing by five instead would describe a population variance if these five values constituted the entire population. State which convention is used.

Other spread measures include quartile deviation and mean absolute deviation. A coefficient of variation expresses standard deviation relative to a suitable nonzero mean, often as a percentage, but is not meaningful for every scale. Percentiles locate observations within an ordered distribution. None of these summaries explains why a deposit formed; geological interpretation still requires context.

Design sampling around the question

Simple random sampling gives eligible units a defined random selection mechanism. Stratified sampling divides the target into meaningful groups and samples within them. Systematic sampling uses regular spacing after an appropriate start, but can align badly with a repeating pattern. Closely spaced geological measurements may be correlated, so a large count does not automatically supply many independent observations.

Design a hypothetical study comparing grain size in channel and overbank deposits. Record the target population, selection rule, sample size by setting and measurement procedure before seeing results. Your final report should include a distribution plot, centre and spread, and a limitation about representation. This produces a defensible description while keeping broader inference proportional to the design.

Self-check

What is the median of 2, 3, 3, 4 and 8?

The median is 3 centimetres.

Why divide by n − 1 for sample variance?

It is the standard correction used when estimating population variance from a sample around its estimated mean.

Can a large convenience sample guarantee representation?

No. Selection bias can remain even when many observations are collected.

Sources and further study

Topic framework: the supplied University of Peshawar BS Geology curriculum, session 2004–05 onward. This independently authored learning resource is not affiliated with the university and does not confer a degree or accreditation.

Chapter 21 multiple choice questions

Choose one best answer for each question. Open each answer panel after making your choice.

For pebble diameters 2, 3, 3, 4 and 8 cm, which pair gives the arithmetic mean and median?
1. Mean 3 cm and median 4 cm
2. Mean 4 cm and median 4 cm
3. Mean 4 cm and median 3 cm
4. Mean 3 cm and median 3 cm
Answer and explanation
C. Mean 4 cm and median 3 cm The sum is 20, so the mean is 4; the middle ordered observation is 3.
The squared deviations from the mean sum to 22 cm² for five sampled diameters. What is the sample variance using n − 1?
1. 5.5 cm²
2. 4.4 cm²
3. 22 cm²
4. 2.35 cm²
Answer and explanation
A. 5.5 cm² Sample variance is 22/(5 − 1) = 5.5 cm²; approximately 2.35 cm is the sample standard deviation.
A researcher divides a formation into channel and overbank deposits, then randomly samples within each group. Which design is being used?
1. A single undifferentiated convenience sample
2. A census of every possible grain
3. Sampling only the most unusual visible pebbles
4. Stratified sampling
Answer and explanation
D. Stratified sampling Stratified sampling separates the target into defined groups and samples within them.
Many measurements are collected along one convenient roadside exposure. What limitation can remain despite the large sample count?
1. Large sample count guarantees complete regional representation.
2. The selection may not represent the full formation.
3. The measurements must all be independent because they are numerous.
4. Increasing sample count automatically removes access bias.
Answer and explanation
B. The selection may not represent the full formation. A large convenience sample can retain selection bias and spatial dependence.
A histogram changes noticeably when class boundaries are changed. What practice supports a transparent analysis?
1. Retain the ungrouped measurements and document bin choices.
2. Keep only the most attractive histogram and discard the values.
3. Treat each binning choice as a different physical rock unit.
4. Assume the altered appearance proves a new geological event.
Answer and explanation
A. Retain the ungrouped measurements and document bin choices. Grouping can affect a display, so original values and bin definitions should remain available.

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