Grain Size Statistics
Measuring grain size produces large tables of weight data showing how much sediment falls into each size class - data that are too raw to use directly for comparison or environmental interpretation. These measurements must be reduced to a more condensed, summary form: tables of raw sieve weights are converted to weight percentages, and from these percentages a small set of statistical parameters - mean size, sorting, skewness, kurtosis - are derived that together describe the grain-size distribution mathematically. [1] Both graphical and mathematical routes are used, and each has distinct advantages.
Graphical Methods
Histogram and Frequency Curve
The simplest graphical representation is the histogram: grain size in phi units is plotted along the horizontal axis, individual weight percent along the vertical axis, and the data appear as a series of bars, one for each sieve interval. [1] Histograms are quick to construct and give an immediate visual impression of the approximate average grain size and sorting - the spread of bar heights around the tallest bar. [1] They have significant limitations, however: the shape of a histogram changes depending on the sieve interval chosen, making histograms from different studies with different sieve spacings difficult to compare directly, and they cannot be used to derive mathematical statistical values. [1]
A frequency curve is essentially a smoothed version of the histogram - a continuous curve drawn through the midpoints of each bar that approximates the shape the histogram would have if infinitely fine sieve intervals were used. [1] The frequency curve’s highest point corresponds to the modal size - the most abundant grain size in the population - but connecting histogram midpoints does not accurately locate this peak. Precise frequency curves are better derived from cumulative curves by specialised graphical methods. [1]
Cumulative Curve
The cumulative curve is the most useful of the grain-size plots. [1] It is generated by plotting grain size against the cumulative weight percent - that is, the percentage of the total sample that is finer than each successive size. Its shape is virtually independent of the sieve interval used, which makes it far more reproducible and comparable between studies than a histogram. [1] Crucially, specific percentile values can be read directly from the curve and used to calculate statistical parameters.
When phi-size data are plotted against an arithmetic ordinate, the cumulative curve typically takes on a characteristic S-shape: the central steeper portion reflects the best-sorted fraction of the population, and the gradient of that central slope directly expresses the degree of sorting - a steep slope indicates good sorting, a gently inclined slope indicates poor sorting. [1]
If the same data are instead plotted on log-probability paper (replacing the arithmetic ordinate with a logarithmic probability scale), a population with a perfectly normal distribution will plot as a straight line. [1] Deviations from normality show up clearly as bends or breaks in this line, making the log-probability plot a sensitive way to detect mixing of grain-size subpopulations. Most natural grain populations in siliciclastic sediments are not normally distributed, which means such deviations are the rule rather than the exception. [1]
Statistical Parameters
Mode, Median, and Mean
Three measures of average grain size are in common use, and they will only coincide when the grain-size distribution is perfectly symmetrical. [1]
The mode is the most frequently occurring grain size - the diameter corresponding to the highest point on a frequency curve or the steepest point (inflection point) on a cumulative curve. [1] Many siliciclastic sediments have a single mode, but bimodal distributions - with separate peaks at a coarse size and a fine size - are not uncommon, and polymodal distributions exist as well. [1]
The median is the midpoint of the distribution by weight: half the grains are coarser and half are finer. It corresponds to the 50th percentile on the cumulative curve. [1]
Mz = (φ16 + φ50 + φ84) / 3 [1]
Sorting and Standard Deviation
Sorting describes the spread of grain sizes around the mean - how tightly the grains cluster around a central value. A well-sorted sediment has grains of nearly uniform size; a poorly sorted one spans a wide range. [1] Mathematically, sorting is expressed as the standard deviation. In a normally distributed population, one standard deviation encompasses the central 68 percent of the area under the frequency curve - that is, 68 percent of the grains fall within plus or minus one standard deviation of the mean size. [1]
The inclusive graphic standard deviation formula is: [1]
σi = (φ84 - φ16) / 4 + (φ95 - φ5) / 6.6 [1]
Standard deviation is expressed in phi units, and the following verbal classes are used: [1]
| Standard Deviation (φ) | Verbal Class | | ---------------------- | ----------------------- | | <0.35 | Very well sorted | | 0.35-0.50 | Well sorted | | 0.50-0.71 | Moderately well sorted | | 0.71-1.00 | Moderately sorted | | 1.00-2.00 | Poorly sorted | | 2.00-4.00 | Very poorly sorted | | >4.00 | Extremely poorly sorted | [1] |
Skewness
Most natural grain-size populations are not symmetrical. A distribution that is skewed has a tail of excess grains extending toward either the fine or coarse end of the spectrum, causing the mean, median, and mode to fall at different values. [1]
Populations with a tail of excess fine particles are positively skewed (fine skewed) - skewed toward higher phi values, meaning toward finer sizes. Populations with a tail of excess coarse particles are negatively skewed (coarse skewed). [1] Skewness thus reveals information about sorting at the extremes of the distribution - whether the deposit contains an extra tail of winnowed fines or a lag of coarser material - which can be informative about transport history.
The inclusive graphic skewness formula is: [1]
SKi = (φ84 + φ16 - 2φ50) / 2(φ84 - φ16) + (φ95 + φ5 - 2φ50) / 2(φ95 - φ5) [1]
Verbal skewness classes are: [1]
| Skewness Value | Verbal Class | | -------------- | ---------------------- | | >+0.30 | Strongly fine skewed | | +0.10 to +0.30 | Fine skewed | | -0.10 to +0.10 | Near symmetrical | | -0.30 to -0.10 | Coarse skewed | | <-0.30 | Strongly coarse skewed | [1] |
Kurtosis
Kurtosis measures how peaked or flat-topped the grain-size frequency curve is relative to a normal distribution. [1] The graphic kurtosis formula is:
KG = (φ95 - φ5) / 2.44(φ75 - φ25) [1]
Although kurtosis is routinely calculated alongside the other grain-size parameters, its geological meaning is not well established and it appears to have limited value in environmental interpretation studies. [1]
The Method of Moments
An alternative route to these same statistical parameters is the method of moments, which computes mean, standard deviation, skewness, and kurtosis mathematically from the raw weight-frequency data without first constructing a graphical cumulative curve. [1] Each moment is calculated by multiplying a weight (the weight frequency in percent) by a distance (the deviation of each size class midpoint from the mean), and summing across all classes. [1] The four formulas are:
Mean (1st moment): x̄φ = Σ(f · m) / n [1]
Standard deviation (2nd moment): σφ = √[Σf(m - x̄φ)² / 100] [1]
Skewness (3rd moment): Skφ = Σf(m - x̄φ)³ / (100 · σφ³) [1]
Kurtosis (4th moment): Kφ = Σf(m - x̄φ)⁴ / (100 · σφ⁴) [1]
where n is the total number in the sample (100 when f is in percent), m is the midpoint of each grain-size class in phi values, and f is the weight percent in each grain-size class present. [1] Moment statistics use all data points rather than only a few percentile readings, so they are in principle more precise than graphical methods. The advent of computers removed the former computational burden, and moment statistics are now in common use. [1]
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References & Citations
- 1.Principles of Sedimentology and Stratigraphy Boggs, Sam Jr.
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