Visualising Distributions

import pandas as pd
import numpy as np

# 20 students — four numeric columns to explore
df = pd.DataFrame({
    'student_id':    range(1, 21),
    'completion_pct':[12, 85, 91, 7, 88, 14, 82, 95, 9, 87,
                      11, 90, 6, 84, 93, 10, 86, 8, 89, 92],
    'quiz_score':    [45, 78, 82, 41, 79, 48, 80, 88, 43, 76,
                      44, 85, 40, 77, 84, 46, 81, 42, 83, 86],
    'days_active':   [2, 18, 21, 1, 19, 3, 17, 24, 2, 20,
                      2, 22, 1, 18, 23, 3, 19, 1, 21, 22],
    'videos_watched':[3, 24, 27, 2, 25, 4, 23, 30, 2, 26,
                      3, 28, 1, 24, 29, 3, 25, 2, 27, 28]
})

# Quick look at the shape of each column
print(df[['completion_pct','quiz_score','days_active','videos_watched']].describe().round(1))

What just happened?

Notice something odd in completion_pct: the mean is 56.1% but the median (50%) is 85.5%. That huge gap is a massive red flag — the mean is being dragged down by something unusual. The standard deviation (39.1) is enormous relative to the range. There are clearly two very different groups in this data. Let's visualise it and see.

def text_histogram(series, col_name, n_bins=8):
    """Prints a simple text histogram so you can see the distribution shape."""
    # pd.cut() divides the values into equal-width buckets
    buckets = pd.cut(series, bins=n_bins)
    counts  = buckets.value_counts().sort_index()

    print(f"Histogram: {col_name}  (n={len(series)})\n")
    max_count = counts.max()

    for bucket, count in counts.items():
        # Scale the bar to fit within 30 characters wide
        bar_len = int(count / max_count * 30)
        bar     = '█' * bar_len
        # Show the range of this bucket and how many students are in it
        print(f"  {str(bucket):<22}  {bar:<30}  {count}")
    print()

text_histogram(df['completion_pct'], 'completion_pct')

What just happened?

pandas' pd.cut() divides the column into 8 equal-width buckets. .value_counts().sort_index() counts how many students fall into each one. The text chart makes the pattern impossible to miss: 10 students are stuck below 17%, and a completely separate cluster of 10 students are above 75% — with nobody in between. This is a textbook bimodal distribution, and the mean of 56% is a completely meaningless number because it represents nobody's actual experience.

Course Completion % — Distribution

# Split students into two groups based on completion
# "Droppers" completed less than 20%
# "Completers" completed more than 75%
droppers    = df[df['completion_pct'] < 20]
completers  = df[df['completion_pct'] > 75]

print(f"Droppers:   {len(droppers)} students")
print(f"Completers: {len(completers)} students")
print()

# Compare the two groups on quiz score and days active
for col in ['quiz_score', 'days_active', 'videos_watched']:
    d_mean = droppers[col].mean()
    c_mean = completers[col].mean()
    d_med  = droppers[col].median()
    c_med  = completers[col].median()

    print(f"--- {col} ---")
    print(f"  Droppers:   mean={d_mean:.1f}  median={d_med:.1f}")
    print(f"  Completers: mean={c_mean:.1f}  median={c_med:.1f}")
    print(f"  Gap: completers score {c_mean - d_mean:+.1f} higher on average")
    print()

What just happened?

pandas' boolean filtering — df[df['completion_pct'] < 20] — selects only the rows where that condition is true. We created two sub-DataFrames (droppers and completers) and ran mean/median on each.

The gaps are enormous. Completers average 82 on quizzes vs 44 for droppers — nearly double. They're active for 21 days vs 2. They watch 26 videos vs 3. These aren't subtle differences — they're two completely different types of student. The product team now knows exactly what to investigate: why do some students drop so early? What happens in the first 2 days?

# Check specific percentiles for quiz scores
# "What score marks the bottom 10%? Bottom 25%? Median? Top 10%?"
percentiles = [10, 25, 50, 75, 90]

print("Quiz Score Percentiles:\n")
for p in percentiles:
    value = df['quiz_score'].quantile(p / 100)   # .quantile() takes a value from 0.0 to 1.0
    # Plain English: "p% of students scored BELOW this value"
    print(f"  {p:>3}th percentile:  {value:.0f} points  "
          f"({p}% of students scored below this)")

print()
# Key business questions answered from percentiles
below_50 = (df['completion_pct'] < 50).sum()
print(f"Students who completed less than half the course: {below_50} out of {len(df)} ({below_50/len(df)*100:.0f}%)")

What just happened?

pandas' .quantile() takes a value between 0.0 and 1.0 — so .quantile(0.25) gives the 25th percentile. We multiply p by 0.01 to convert from "25" to "0.25."

The quiz scores confirm the bimodal story: the 25th percentile is 43 and the 50th is 79 — a huge jump of 36 points in one step. That's not a smooth curve; there are two separate clusters. And 50% of students completed less than half the course — a damning metric for the product team.

def distribution_report(dataframe, columns):
    """Diagnoses the shape of each column and recommends an action."""
    for col in columns:
        s      = dataframe[col].dropna()
        mean   = s.mean()
        median = s.median()
        skew   = s.skew()
        p10    = s.quantile(0.10)
        p90    = s.quantile(0.90)

        # Diagnose shape based on skewness
        if skew > 1:
            shape  = "Right-skewed ↗"
            action = "Consider log transform before modelling"
        elif skew < -1:
            shape  = "Left-skewed ↙"
            action = "Review low-end values — may need transformation"
        elif abs(mean - median) > median * 0.3:
            shape  = "Bimodal or heavy outliers ⚡"
            action = "Split into subgroups and analyse separately"
        else:
            shape  = "Roughly symmetric ✓"
            action = "Safe to use as-is"

        print(f"{'='*50}")
        print(f"  Column: {col}")
        print(f"  Shape:  {shape}")
        print(f"  Mean={mean:.1f}  Median={median:.1f}  Skew={skew:.2f}")
        print(f"  10th pct={p10:.1f}  →  90th pct={p90:.1f}")
        print(f"  Action: {action}")
    print(f"{'='*50}")

distribution_report(df, ['completion_pct','quiz_score','days_active','videos_watched'])

What just happened?

pandas' .skew(), .quantile(), .mean(), and .median() all run inside one function that packages the results into a readable report. The shape diagnosis is based on skewness and the mean-median gap — two simple checks that catch most distribution problems.

Every single column flags as bimodal and recommends splitting into subgroups. That's the right call — this data describes two fundamentally different types of student, and any model trained on the combined data would be confused from the start. The distribution report just saved the modelling team from a very expensive mistake.

Teacher's Note

Never report a mean without checking if the distribution is unimodal first. The mean of 56% completion we saw here is technically correct — but it's completely misleading. It makes the course sound mediocre, when actually it has two extreme groups: students who love it and finish almost everything, and students who drop within the first two days.

In Python, you'd use matplotlib.pyplot.hist() or seaborn.histplot() to create real visual histograms. The text histogram and HTML charts in this lesson show you the same shape — the code for real charts is almost identical, just with plt.show() at the end. We'll use full seaborn visualisations in Lesson 29 onwards.

Up Next · Lesson 27

Visualising Relationships

Scatter plots, line charts, and pair plots — how to see the connections between two variables before you ever run a model on them.