Day 2: Your Data at a Glance — Descriptive Statistics

The Problem

Dr. Sarah Chen stares at her screen. The Illumina NovaSeq 6000 finished its run overnight, and now she has 10,247 quality scores — one for each tile on the flow cell. Her PI needs a decision by the morning meeting: is this data usable, or do they need to re-run the library, burning another $4,000 and two days?

She cannot read 10,247 numbers. She cannot scroll through them and develop an intuition. She has five minutes before the meeting starts. What she needs is a way to compress 10,247 numbers into a handful of meaningful summaries that answer three questions: What is the typical quality? How much does it vary? Are there any red flags?

This is the job of descriptive statistics. They are the first thing you compute, every time, before you run a single test. Get them wrong — or skip them — and everything downstream is built on sand.

What Are Descriptive Statistics?

Descriptive statistics are summaries. Think of them as a movie trailer for your data. The full movie (the raw dataset) might be two hours long, but the trailer gives you the genre, the tone, and the key plot points in two minutes. A good trailer does not lie about the movie. A good set of descriptive statistics does not lie about the data.

There are three things you need to know about any dataset:

1. Center — Where is the “middle” of the data?
2. Spread — How far do values range from that center?
3. Shape — Is the data symmetric? Skewed? Are there outliers?

Why Do These Three Matter?

Center tells you what is “normal.” If the mean quality score of your sequencing run is Q35, you know tiles are performing well. If it is Q15, something went wrong. Without center, you have no reference point — you cannot say whether a single observation is typical or alarming. In clinical genomics, center defines the baseline: what is the typical variant allele frequency in this cohort? What is the average coverage across your target regions?

Spread tells you how much you can trust the center. Two experiments might both report a mean IC50 of 12 nM, but one has values ranging from 11 to 13 (tight, reproducible) while the other ranges from 2 to 45 (noisy, unreliable). The center is the same — the spread tells you completely different stories. High spread means your next measurement could land anywhere; low spread means you can make confident predictions. In RNA-seq, high within-group variance buries real differential expression in noise.

Shape tells you which statistical tools are safe to use. Almost every standard statistical test (t-test, ANOVA, linear regression) assumes the data is approximately bell-shaped (normally distributed). If your data is heavily right-skewed — which gene expression nearly always is — those tests give wrong answers. Shape also reveals hidden structure: a bimodal distribution might mean you have two distinct populations mixed together (e.g., responders and non-responders to a drug). Ignoring shape is the single most common reason published biomedical results fail to replicate.

Key insight: Center without spread is meaningless (“the average temperature is 72°F” tells you nothing if the range is -20 to 160). Spread without shape is dangerous (a standard deviation assumes symmetry, but skewed data violates this). You always need all three.

Every statistical analysis starts here. If you skip descriptive statistics and jump straight to hypothesis testing, you are performing surgery without an examination.

Measures of Center

Mean (Arithmetic Average)

The mean is the balance point. If your data values were weights placed along a ruler, the mean is the spot where the ruler would balance perfectly.

Formula: x̄ = (1/n) ∑ xᵢ

The mean uses every data point, which is both its strength and its weakness. It is the most efficient estimator of center when data is symmetric with no outliers. But it is exquisitely sensitive to extreme values.

Example: Five gene expression values (FPKM): 12, 15, 14, 13, 16. Mean = 14.0. Reasonable.

Now add one highly expressed gene: 12, 15, 14, 13, 16, 5000. Mean = 845.0. The mean has been dragged from 14 to 845 by a single outlier. It no longer represents “typical” expression.

Common pitfall: In genomics, gene expression distributions are heavily right-skewed. Reporting mean FPKM/TPM values without acknowledging this skew is misleading. The median is almost always a better summary for expression data.

Median (Middle Value)

The median is the value that splits the data in half: 50% of observations fall below it, 50% above. Sort the data and pick the middle number (or average the two middle numbers if n is even).

For our outlier-contaminated expression data: sorted = 12, 13, 14, 15, 16, 5000. Median = (14 + 15) / 2 = 14.5. The outlier barely matters.

The median is robust — it resists the pull of extreme values. This makes it the preferred measure of center for skewed distributions, which are the norm in biology.

Mode (Most Frequent Value)

The mode is the most common value. It is most useful for categorical or discrete data: the most common blood type, the most frequent variant allele, the peak of a histogram.

For continuous data, the mode is the peak of the density curve. Bimodal distributions (two peaks) arise in biology more often than you might expect — for instance, CpG methylation levels often cluster near 0% and 100%, reflecting fully unmethylated and fully methylated states.

When to Use Each

Key insight: Always report both mean and median. If they differ substantially, your data is skewed, and the median is the more honest summary.

Measures of Spread

Knowing the center is not enough. Two datasets can have identical means and wildly different behaviors. Consider drug response in two patient cohorts — both might have a mean survival of 12 months, but in one cohort everyone lives 11-13 months, while in the other, half die in 2 months and half live 22 months. The clinical implications are completely different.

Range

The simplest measure: maximum minus minimum. It tells you the total extent of the data but is completely determined by the two most extreme points.

Variance and Standard Deviation

Variance measures the average squared distance from the mean:

Variance: s² = (1 / (n-1)) ∑ (xᵢ - x̄)²

Standard Deviation: s = √(s²)

We divide by (n-1) rather than n (Bessel’s correction) because a sample underestimates the true population variance. The standard deviation is in the same units as the data, making it more interpretable than variance.

Rule of thumb: For normally distributed data, about 68% of values fall within 1 SD of the mean, 95% within 2 SDs, and 99.7% within 3 SDs. If a quality score is more than 3 SDs below the mean, something is wrong with that tile.

Interquartile Range (IQR)

The IQR is the range of the middle 50% of the data: Q3 - Q1, where Q1 is the 25th percentile and Q3 is the 75th percentile.

Like the median, the IQR is robust to outliers. It is the foundation of the box plot and a standard measure of spread for skewed data.

Coefficient of Variation (CV)

CV = (SD / Mean) × 100%. The CV expresses variability relative to the mean, making it useful for comparing spread across measurements with different scales.

Example: If gene A has mean expression 1000 TPM with SD 100, and gene B has mean 10 TPM with SD 5, which is more variable? Gene A has higher SD, but gene B has a higher CV (50% vs 10%). Gene B’s expression is relatively more variable.

Shape: Skewness and Kurtosis

Skewness

Skewness measures asymmetry. A skewness of 0 means perfectly symmetric (like the normal distribution). Positive skewness means a right tail (most values cluster low, with a few very high values). Negative skewness means a left tail.

Biological reality: Most biological measurements are positively skewed. Gene expression, protein abundance, cell counts, read depths — all tend to have many low values and a few very high ones. This is because multiplicative processes (gene regulation cascades, exponential growth) naturally produce right-skewed distributions.

Kurtosis

Kurtosis measures the “tailedness” of the distribution — how likely extreme values are compared to a normal distribution. High kurtosis means heavy tails (more outliers than expected). Low kurtosis means light tails.

In genomics, variant allele frequency distributions often have high kurtosis, reflecting the mixture of common variants (clustered near 50%) and rare variants (near 0%).

Key insight: If skewness is far from 0 or kurtosis is far from 3, your data is not normally distributed, and parametric tests that assume normality may give misleading results. We will explore this deeply on Day 3.

Descriptive Statistics in BioLang

Let us return to Dr. Chen’s problem. She has 10,247 quality scores and five minutes.

Loading and Summarizing Data

set_seed(42)
# Simulate sequencing quality scores (Phred scale, typically 0-40)
let quality_scores = rnorm(10247, 32.5, 3.8)

# One-line comprehensive summary
let stats = summary(quality_scores)
print(stats)
# Output:
#   n:        10247
#   mean:     32.48
#   median:   32.51
#   sd:       3.81
#   min:      17.23
#   max:      47.12
#   q1:       29.93
#   q3:       35.07
#   iqr:      5.14
#   skewness: -0.01
#   kurtosis: 2.99

That single call answers Dr. Chen’s three questions immediately. Mean quality is 32.5 (good — above the Phred 30 threshold). The SD is 3.8 (moderate spread). Skewness is near zero (symmetric). No red flags. The run is usable.

Computing Individual Statistics

# Individual measures of center
let avg = mean(quality_scores)       # 32.48
let med = median(quality_scores)     # 32.51
let mod = mode(quality_scores)       # 32 (rounded to nearest integer)

# Individual measures of spread
let sd = stdev(quality_scores)       # 3.81
let v = variance(quality_scores)     # 14.52
let r = range_stat(quality_scores)   # [17.23, 47.12]
let q = quantile(quality_scores, [0.25, 0.5, 0.75])  # [29.93, 32.51, 35.07]
let i = iqr(quality_scores)          # 5.14

# Shape
let sk = skewness(quality_scores)    # -0.01
let ku = kurtosis(quality_scores)    # 2.99

print("Mean: {avg}, Median: {med}")
print("SD: {sd}, IQR: {i}")
print("Skewness: {sk}, Kurtosis: {ku}")

The summary() Function

For a more detailed report:

let report = summary(quality_scores)
print(report)
# Output:
#   Variable:    quality_scores
#   Count:       10247
#   Mean:        32.48
#   Std Dev:     3.81
#   Min:         17.23
#   25%:         29.93
#   50%:         32.51
#   75%:         35.07
#   Max:         47.12
#   Skewness:    -0.01
#   Kurtosis:    2.99
#   CV:          11.73%
#   SE(mean):    0.038

Working with Real Sequencing Data

set_seed(42)
# In practice, you'd load from a file:
# let scores = read_csv("data/expression.csv") |> column("phred_score")

# Simulate a problematic run with bimodal quality
let good_tiles = rnorm(8000, 33.0, 2.5)
let bad_tiles = rnorm(2247, 18.0, 4.0)
let mixed_scores = good_tiles + bad_tiles

let mixed_stats = summary(mixed_scores)
print(mixed_stats)
# Mean: 29.7 — looks okay at first glance
# Median: 32.1 — the median reveals the truth: most tiles are fine
# Skewness: -1.4 — strongly left-skewed, warning sign!

# The mean alone would have hidden the problem.
# Always look at the full distribution.

Common pitfall: Relying on the mean alone can hide bimodal distributions. The “bad tile” problem above is common in sequencing — a mean of 29.7 looks passable, but 22% of tiles are failing. Always visualize.

Visualization: Always Plot Before You Test

Numbers tell part of the story. Plots tell the rest. Anscombe’s Quartet — four datasets with identical means, variances, and correlations but wildly different structures — demonstrates why you must always look at your data.

Histograms

A histogram bins your data and counts frequencies. It reveals the distribution’s shape at a glance.

# Basic histogram (uses quality_scores from block above — click "Run All Above + This")
histogram(quality_scores, {bins: 50, title: "Sequencing Quality Distribution"})

# Histogram of the problematic run (uses mixed_scores from earlier block)
histogram(mixed_scores, {bins: 50, title: "Bimodal Quality — Problem Run"})
# This immediately reveals the two peaks that the mean masked.

The good run produces a single symmetric peak around 32-33. The problem run shows two distinct peaks — one centered at 18 and one at 33. No summary statistic captures this as effectively as the histogram.

Box Plots

A box plot displays the median (center line), IQR (box), and whiskers (typically 1.5 × IQR). Points beyond the whiskers are marked as individual outliers.

set_seed(42)
# Compare quality across lanes
let lane1 = rnorm(2000, 33.0, 2.0)
let lane2 = rnorm(2000, 31.5, 3.5)
let lane3 = rnorm(2000, 28.0, 5.0)

let bp_table = table({"Lane 1": lane1, "Lane 2": lane2, "Lane 3": lane3})
boxplot(bp_table, {title: "Quality Score Distribution by Lane"})
# Lane 3 immediately stands out: lower median, wider spread, more outliers.

Box plots shine when comparing groups. You can see at a glance which lane is problematic, without computing a single number.

Combining Plots and Stats

set_seed(42)
# Full QC report in a few lines
let scores = rnorm(10000, 32.5, 3.8)

# Side-by-side: histogram + box plot
histogram(scores, {bins: 40, title: "Quality Score Distribution"})
boxplot(scores, {title: "Quality Score Box Plot"})

# Summary table
let stats = summary(scores)
print("QC Decision: {if stats.mean > 30 then 'PASS' else 'FAIL'}")
print("Mean Q-score: {stats.mean:.1}")
print("Tiles below Q20: {scores |> filter(|s| s < 20) |> len()}")

Python and R Equivalents

For those coming from Python or R, here are the equivalent operations.

Python:

import numpy as np
from scipy import stats

scores = np.random.normal(32.5, 3.8, 10247)

# Measures of center
np.mean(scores)          # 32.48
np.median(scores)        # 32.51
stats.mode(scores)       # most frequent

# Measures of spread
np.std(scores, ddof=1)   # 3.81 (ddof=1 for sample SD)
np.var(scores, ddof=1)   # 14.52
np.percentile(scores, [25, 50, 75])
stats.iqr(scores)        # 5.14

# Shape
stats.skew(scores)       # -0.01
stats.kurtosis(scores)   # -0.01 (scipy uses excess kurtosis)

# Comprehensive summary
import pandas as pd
pd.Series(scores).describe()

R:

scores <- rnorm(10247, mean = 32.5, sd = 3.8)

# Measures of center
mean(scores)       # 32.48
median(scores)     # 32.51

# Measures of spread
sd(scores)         # 3.81
var(scores)        # 14.52
quantile(scores, c(0.25, 0.5, 0.75))
IQR(scores)        # 5.14

# Shape (requires moments package)
library(moments)
skewness(scores)   # -0.01
kurtosis(scores)   # 2.99

# Comprehensive summary
summary(scores)

# Visualization
hist(scores, breaks = 50, main = "Quality Distribution")
boxplot(scores, main = "Quality Box Plot")

Worked Example: The QC Decision

Let us walk through Dr. Chen’s complete analysis.

set_seed(42)
# Step 1: Load the data
let scores = rnorm(10247, 32.5, 3.8)

# Step 2: Quick summary
let stats = summary(scores)

# Step 3: QC criteria
let pass_threshold = 30.0     # Minimum acceptable mean quality
let fail_fraction_limit = 0.10 # Max 10% tiles below Q20

# Step 4: Compute QC metrics
let tiles_below_q20 = scores |> filter(|s| s < 20) |> len()
let fraction_below_q20 = tiles_below_q20 / len(scores)

# Step 5: Decision
let qc_pass = stats.mean >= pass_threshold and fraction_below_q20 <= fail_fraction_limit

print("=== Sequencing Run QC Report ===")
print("Total tiles:          {len(scores)}")
print("Mean quality:         {stats.mean:.2}")
print("Median quality:       {stats.median:.2}")
print("Std deviation:        {stats.sd:.2}")
print("Tiles below Q20:      {tiles_below_q20} ({fraction_below_q20 * 100:.1}%)")
print("QC Result:            {if qc_pass then 'PASS' else 'FAIL'}")
print("================================")

# Step 6: Visualize
histogram(scores, {bins: 50, title: "Run QC: Quality Score Distribution"})

This takes about 10 seconds to run. Dr. Chen has her answer well before the meeting.

Exercises

Exercise 1: Gene Expression Summary

Compute descriptive statistics for a set of gene expression values and identify the best measure of center.

# Gene expression values (TPM) for 20 genes
let expression = [0.5, 1.2, 3.4, 5.1, 8.7, 12.3, 15.0, 22.4, 45.6, 78.9,
                  120.5, 250.3, 0.1, 2.8, 6.5, 0.3, 1100.0, 33.2, 0.8, 18.5]

# TODO: Compute mean, median, stdev, skewness
# TODO: Which measure of center best represents "typical" expression? Why?
# TODO: Create a histogram. What shape do you see?

Exercise 2: Comparing Sequencing Runs

Two sequencing runs produced quality scores. Determine which run is better.

set_seed(42)
let run_a = rnorm(5000, 31.0, 2.5)
let run_b = rnorm(5000, 33.0, 6.0)

# TODO: Compute summary() for both runs
# TODO: Which has higher mean? Which has lower variability?
# TODO: Compute the CV for each. Which is more consistent?
# TODO: Create side-by-side box plots
# TODO: If you could only pick one run, which would you choose and why?

Exercise 3: Outlier Detection

Identify outliers using the IQR method (values below Q1 - 1.5IQR or above Q3 + 1.5IQR).

# Protein abundance measurements with some suspicious values
let protein = [45.2, 48.1, 50.3, 47.8, 49.5, 46.7, 51.2, 48.9,
               150.0, 47.3, 49.8, 46.1, 50.5, 48.4, 3.0, 49.1]

# TODO: Compute Q1, Q3, IQR
# TODO: Calculate lower and upper fences
# TODO: Identify which values are outliers
# TODO: Compute mean with and without outliers — how much does it change?

Exercise 4: Multi-Sample QC Dashboard

Build a QC summary for multiple samples.

set_seed(42)
let samples = {
    "Sample_A": rnorm(1000, 34.0, 2.0),
    "Sample_B": rnorm(1000, 29.0, 4.0),
    "Sample_C": rnorm(1000, 32.0, 3.0),
    "Sample_D": rnorm(1000, 20.0, 5.0)
}

# TODO: Loop through samples, compute summary() for each
# TODO: Flag any sample with mean < 25 or CV > 20%
# TODO: Create a box plot comparing all four samples

Key Takeaways

• Descriptive statistics compress large datasets into interpretable summaries of center, spread, and shape.
• The mean is efficient but outlier-sensitive; the median is robust. For skewed biological data, prefer the median.
• Standard deviation measures absolute spread; IQR is robust to outliers; CV enables comparison across different scales.
• Skewness and kurtosis reveal whether your data’s shape matches the assumptions of common statistical tests.
• Always visualize your data with histograms and box plots before computing any test. Summary statistics can hide multimodal distributions, outliers, and other structural features.
• summary() in BioLang provides comprehensive descriptive statistics in a single function call.
• Descriptive statistics are not optional preliminaries — they are the foundation of every analysis.

What’s Next

You now know how to summarize data, but you may have noticed something: we keep saying “normally distributed” and “skewed” without precisely defining what a distribution is. Tomorrow, on Day 3, we dive into the mathematical shapes that biological data follows — the normal distribution, the log-normal, the Poisson, and the binomial. You will learn why gene expression data refuses to be normal, why read counts follow a Poisson process (sort of), and how to test whether your data fits the distribution you think it does. Understanding distributions is the key to choosing the right statistical test — and avoiding the wrong one.