Descriptive statistics

A breeding programme may record body weight, disease score, or harvest yield for hundreds of individuals. Before fitting a genomic prediction model, the breeder needs to know what values are typical, how much individuals differ, and whether a few unusual records deserve investigation. Descriptive statistics answer these questions by reducing a collection of observations to numerical and graphical summaries. They describe the recorded sample; by themselves, they do not explain why the observations differ or predict an unobserved individual.

Centre and spread answer different questions

Consider two hypothetical groups of five selection candidates. Both groups have a mean weight of 10 kg, but their values differ in spread.

Candidate Group A (kg) Group B (kg)
1 8 4
2 9 7
3 10 10
4 11 13
5 12 16

The equal means do not make the groups equivalent. Group A lies close to 10 kg, whereas Group B includes much lighter and heavier candidates. A measure of centre identifies a typical value. A measure of spread describes how far observations tend to lie from that centre. Genomic prediction needs both: marker effects cannot help distinguish candidates if the target trait has no variation, and a centre alone hides the amount of variation available to the model.

Measures of centre

The arithmetic mean of observations \(y_1,\ldots,y_n\) is

\[ \bar y = \frac{1}{n}\sum_{i=1}^{n} y_i, \]

where \(n\) is the number of observations and \(\bar y\) is their sample mean. Every observation contributes its magnitude, so one extreme value can move the mean substantially. This sensitivity is useful when the magnitude of every measurement matters, but it also means that data errors can distort the summary.

The median is the middle value after sorting the observations. For an even number of observations, it is conventionally the mean of the two middle values. Because the median depends on order rather than the distance of every observation from the centre, it changes less than the mean when one observation is extreme.

The mode is the most frequent observed value or category. A sample can have no repeated value, one mode, or several equally frequent modes. For continuously measured traits, exact repetitions may be rare and the mode can depend on measurement precision or histogram bins. It is often more informative for genotype or phenotype categories than for precise continuous measurements.

Measures of position and spread

A quantile divides ordered observations by cumulative proportion. The first quartile \(Q_1\) marks the 25th percentile, the median marks the 50th, and the third quartile \(Q_3\) marks the 75th. Software packages use several interpolation conventions for sample quantiles, so a small sample can yield slightly different values under different conventions.

The range is the maximum minus the minimum. It is easy to interpret but uses only two observations. The interquartile range,

\[ \operatorname{IQR}=Q_3-Q_1, \]

describes the width of the middle half of the data. It is less sensitive than the range to observations in the tails.

Variance begins with a deviation, \(y_i-\bar y\), the signed distance between one observation and the mean. Deviations sum to zero, so simply averaging them would report no spread. Squaring prevents positive and negative deviations from cancelling. The sample variance is

\[ s^2=\frac{1}{n-1}\sum_{i=1}^{n}(y_i-\bar y)^2. \]

Variance has squared units. If weight is recorded in kilograms, its variance is measured in square kilograms. The sample standard deviation is \(s=\sqrt{s^2}\), which restores the original measurement unit.

Assume \(Y_1,\ldots,Y_n\) are independent and identically distributed with finite mean \(\mu\) and variance \(\sigma^2\). Estimating the mean from the same observations forces the deviations from \(\bar Y\) to sum to zero. This is one linear constraint, leaving \(n-1\) degrees of freedom; it does not mean that the individual deviations are independent.

The identity

\[ \sum_{i=1}^{n}(Y_i-\bar Y)^2 =\sum_{i=1}^{n}(Y_i-\mu)^2-n(\bar Y-\mu)^2 \]

makes the correction explicit. Taking expectations gives

\[ E\left[\sum_{i=1}^{n}(Y_i-\bar Y)^2\right] =n\sigma^2-n\operatorname{Var}(\bar Y) =n\sigma^2-n\left(\frac{\sigma^2}{n}\right) =(n-1)\sigma^2. \]

Dividing by \(n-1\) therefore makes \(s^2\) unbiased for \(\sigma^2\) under these assumptions. If the observations are the entire population of interest rather than a random sample from it, division by \(n\) answers a different descriptive question.

Manual calculation with the project data

The recurring dataset in this curriculum is simulated. It contains 200 individuals, their marker genotypes, phenotypes, pedigrees, and true breeding values. For a calculation that can be checked by hand, take the first five continuous phenotypes and round them to whole units. Rounding is used only for this teaching calculation; applied summaries retain the raw values.

Code
d <- readRDS("../../demo-data/main/demo_data.rds")
y <- round(d$pheno$y_cont_qtl_snp[1:5])
y
[1] -1 -3 -2  1 -2

The values are \((-1,-3,-2,1,-2)\). Their sum is \(-7\), so the mean is \(-7/5=-1.4\). Sorting gives \((-3,-2,-2,-1,1)\); the middle value and median is \(-2\). The value \(-2\) occurs twice, more often than any other value, so it is the mode.

Subtracting the mean gives the deviations \((0.4,-1.6,-0.6,2.4,-0.6)\). Their squares sum to

\[ 0.4^2+(-1.6)^2+(-0.6)^2+2.4^2+(-0.6)^2=9.2. \]

The sample variance is therefore \(9.2/(5-1)=2.3\), and the sample standard deviation is \(\sqrt{2.3}=1.517\) units. Under R’s default quantile convention, \(Q_1=-2\) and \(Q_3=-1\), giving an IQR of 1. The range is \(1-(-3)=4\).

Code
mode_value <- as.numeric(names(which.max(table(y))))
c(
  mean = mean(y),
  median = median(y),
  mode = mode_value,
  variance = var(y),
  sd = sd(y),
  range = diff(range(y)),
  iqr = IQR(y)
)
     mean    median      mode  variance        sd     range       iqr 
-1.400000 -2.000000 -2.000000  2.300000  1.516575  4.000000  1.000000

These summaries describe five rounded observations, not the full simulated population. The negative centre is also relative to the arbitrary scale used by the simulation; it does not mean that an individual has a biologically negative trait. On all 200 raw records, the centre and spread differ because rounding has been removed and many more individuals contribute.

Code
summary(d$pheno$y_cont_qtl_snp)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
-3.4565 -0.5038  0.4788  0.4106  1.2622  3.8064
Code
c(
  variance = var(d$pheno$y_cont_qtl_snp),
  sd = sd(d$pheno$y_cont_qtl_snp)
)
variance       sd 
1.774446 1.332083

For these 200 simulated records, the raw mean is approximately 0.411 and the standard deviation is approximately 1.332. These are realised summaries of one simulated dataset. They are not universal values for a species, trait, or breeding programme.

Outliers require investigation, not automatic deletion

An outlier is an observation that appears unusually distant from most of the data under a stated comparison rule. The common boxplot rule flags values below \(Q_1-1.5\operatorname{IQR}\) or above \(Q_3+1.5\operatorname{IQR}\). It is a screening convention, not proof that an observation is false.

An unusual phenotype could result from a transcription error, a different measurement protocol, disease, environment, or genuine biology. Removing it without checking can discard valuable genetic variation and make later predictions appear more precise than they are. A breeder should examine the record, metadata, and biological context before deciding whether correction, exclusion, or retention is justified.

Exercises

  1. Round rows 6 to 10 of y_cont_qtl_snp. Calculate their mean, median, mode, sample variance, and sample standard deviation. Compare their centre and spread with rows 1 to 5.

The rounded values are \((-2,0,-1,-1,-2)\). Their mean is \(-1.2\), the median is \(-1\), and the two modes are \(-2\) and \(-1\). The sample variance is 0.7, and the sample standard deviation is approximately 0.837. Compared with rows 1 to 5, this group has a higher mean and a smaller standard deviation. The comparison is descriptive and does not establish a genetic cause for the difference.

Code
exercise_y <- round(d$pheno$y_cont_qtl_snp[6:10])
exercise_counts <- table(exercise_y)
exercise_modes <- as.numeric(
  names(exercise_counts[exercise_counts == max(exercise_counts)])
)
c(
  mean = mean(exercise_y),
  median = median(exercise_y),
  variance = var(exercise_y),
  sd = sd(exercise_y)
)
    mean   median variance       sd 
-1.20000 -1.00000  0.70000  0.83666
Code
exercise_modes
[1] -2 -1
  1. A hypothetical trait has values \((5,5,6,7,27)\). Compare the mean and median. Which measure better represents the four observations clustered between 5 and 7?

The mean is \((5+5+6+7+27)/5=10\), while the median is 6. The value 27 pulls the mean above four of the five observations. The median better represents the centre of that cluster, but the value 27 should be investigated rather than deleted automatically.

  1. Two hypothetical groups both have mean 4. Group C is \((3,4,5)\), and Group D is \((0,4,8)\). Calculate their sample variances and explain what the comparison adds beyond their equal means.

For Group C, the squared deviations sum to 2, so the sample variance is \(2/(3-1)=1\). For Group D, they sum to 32, so the sample variance is \(32/(3-1)=16\). The equal means identify the same centre, while the variances show that Group D is much more dispersed.

  1. A sample has \(Q_1=4\) and \(Q_3=10\). Calculate its IQR. Why can this be more informative than the range when one observation is unusually distant?

The IQR is \(10-4=6\). It describes the middle half of the ordered data, so it is less sensitive than the range, which depends only on the minimum and maximum values.

  1. A phenotype is flagged by the boxplot rule. Should it be deleted automatically? State two checks that the article recommends before deciding.

No. The rule is a screening convention, not proof that a record is false. The breeder should examine the record and its metadata or measurement protocol, then consider biological context such as disease, environment, or genuine genetic variation before correction, exclusion, or retention.

Descriptive statistics treat the recorded values as fixed observations. The next step is to describe uncertain outcomes before they are observed. Probability and random variables provide that language and connect Mendelian segregation to the statistical models used in genomic prediction.