Mean, Median, Mode

Probability and Statistics

Easy

Problem

Theory

Central tendency describes where the "center" of a dataset lies. The three most common measures are:

• Mean: The arithmetic average
• Median: The middle value when data is sorted
• Mode: The most frequently occurring value

Each measure has different properties and is appropriate in different situations.

The Mean (Arithmetic Average)

The mean is the sum of all values divided by the count:

\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i = \frac{x_1 + x_2 + ... + x_n}{n}

Properties:

• Uses all data points
• Sensitive to outliers
• Minimizes sum of squared deviations
• Has nice mathematical properties (unbiased estimator of population mean)

Computing the Mean: Example

Data: [4, 8, 6, 5, 3, 9, 7]

Step 1: Sum the values

4 + 8 + 6 + 5 + 3 + 9 + 7 = 42

Step 2: Count the values

n = 7

Step 3: Divide

\bar{x} = \frac{42}{7} = 6

The mean is 6.

The Median

The median is the middle value when data is sorted in order:

For odd n: The median is the value at position \frac{n+1}{2}.

For even n: The median is the average of values at positions \frac{n}{2} and \frac{n}{2} + 1.

Properties:

• Robust to outliers
• Minimizes sum of absolute deviations
• Good for skewed distributions
• 50th percentile

Computing the Median: Odd n

Data: [4, 8, 6, 5, 3, 9, 7]

Step 1: Sort the data

[3, 4, 5, 6, 7, 8, 9]

Step 2: Find the middle position

\text{position} = \frac{n+1}{2} = \frac{7+1}{2} = 4

Step 3: The median is the 4th value

\text{median} = 6

Computing the Median: Even n

Data: [4, 8, 6, 5, 3, 9]

Step 1: Sort the data

[3, 4, 5, 6, 8, 9]

Step 2: Find the two middle positions

\text{positions} = \frac{n}{2} \text{ and } \frac{n}{2} + 1 = 3 \text{ and } 4

Step 3: Average the 3rd and 4th values

\text{median} = \frac{5 + 6}{2} = 5.5

The Mode

The mode is the value that appears most frequently:

Properties:

• Can be used with categorical data
• May not exist (if all values are unique)
• May not be unique (multimodal distributions)
• Not affected by outliers

Computing the Mode: Example 1

Data: [4, 8, 6, 5, 3, 6, 7, 6, 9]

Count occurrences:

• 3 appears 1 time
• 4 appears 1 time
• 5 appears 1 time
• 6 appears 3 times
• 7 appears 1 time
• 8 appears 1 time
• 9 appears 1 time

Mode = 6 (appears most frequently)

Computing the Mode: No Mode

Data: [4, 8, 6, 5, 3, 9, 7]

Each value appears exactly once. There is no mode (or we say all values are modes).

Computing the Mode: Multiple Modes

Data: [4, 8, 6, 5, 3, 6, 8, 9]

Count occurrences:

• 6 appears 2 times
• 8 appears 2 times
• All others appear 1 time

Modes = 6 and 8 (bimodal distribution)

Comparison: Mean vs Median

Symmetric distributions:

• Mean \approx Median
• Either measure works well

Right-skewed distributions (long right tail):

• Mean > Median
• Median is often preferred

Left-skewed distributions (long left tail):

• Mean < Median
• Median is often preferred

With outliers:

• Mean is pulled toward outliers
• Median is robust to outliers

Effect of Outliers: Example

Data without outlier: [10, 12, 11, 13, 12, 11, 14]

Mean = (10+12+11+13+12+11+14)/7 = 83/7 = 11.86

Median = 12 (middle of sorted [10, 11, 11, 12, 12, 13, 14])

Data with outlier: [10, 12, 11, 13, 12, 11, 100]

Mean = (10+12+11+13+12+11+100)/7 = 169/7 = 24.14

Median = 12 (middle of sorted [10, 11, 11, 12, 12, 13, 100])

The outlier dramatically affects the mean but not the median.

When to Use Each Measure

Use Mean when:

• Data is roughly symmetric
• No significant outliers
• You need to use the value in further calculations
• You want to account for all values

Use Median when:

• Data is skewed
• Outliers are present
• Reporting "typical" values (e.g., income, house prices)
• Working with ordinal data

Use Mode when:

• Data is categorical
• You want the most common value
• Describing distribution shape (number of peaks)

Relationship in Different Distributions

Symmetric distribution:

\text{Mean} = \text{Median} = \text{Mode}

Right-skewed distribution:

\text{Mode} < \text{Median} < \text{Mean}

Left-skewed distribution:

\text{Mean} < \text{Median} < \text{Mode}

This relationship is approximate and does not always hold exactly.

Weighted Mean

When data points have different importances (weights):

\bar{x}_w = \frac{\sum_{i=1}^{n} w_i x_i}{\sum_{i=1}^{n} w_i}

Example: Calculate GPA where courses have different credits.

Grades: A(4), B(3), A(4), C(2) Credits: 3, 4, 2, 3

\text{GPA} = \frac{4(3) + 3(4) + 4(2) + 2(3)}{3 + 4 + 2 + 3} = \frac{12 + 12 + 8 + 6}{12} = \frac{38}{12} = 3.17

Trimmed Mean

The trimmed mean removes a percentage of extreme values before computing the mean:

Process:

1. Sort the data
2. Remove the lowest k\% and highest k\%
3. Compute mean of remaining values

Example: 10% trimmed mean of 20 values removes the 2 smallest and 2 largest.

This provides robustness to outliers while still using most data.

Geometric Mean

For positive values, the geometric mean is:

\bar{x}_g = \sqrt[n]{x_1 \cdot x_2 \cdot ... \cdot x_n} = \left(\prod_{i=1}^{n} x_i\right)^{1/n}

Use cases:

• Growth rates and ratios
• Values spanning orders of magnitude
• Proportional data

Example: Growth rates of 10%, 20%, 30%

Geometric mean = \sqrt[3]{1.1 \times 1.2 \times 1.3} = \sqrt[3]{1.716} = 1.197

Average growth rate \approx 19.7\%

Harmonic Mean

The harmonic mean is:

\bar{x}_h = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}}

Use cases:

• Averaging rates (speed, productivity)
• F1-score in machine learning

Example: Drive 60 mph one way, 40 mph return

Harmonic mean = \frac{2}{\frac{1}{60} + \frac{1}{40}} = \frac{2}{\frac{2+3}{120}} = \frac{2 \times 120}{5} = 48 mph

The average speed is 48 mph, not 50 mph.

Relationship Between Means

For positive values:

\text{Harmonic Mean} \leq \text{Geometric Mean} \leq \text{Arithmetic Mean}

Equality holds only when all values are equal.

Central Tendency in Machine Learning

Imputation:

• Missing values often imputed with mean or median
• Median is preferred for skewed features

Loss functions:

• Mean squared error is minimized by the mean
• Mean absolute error is minimized by the median

Baseline models:

• Predicting the mean (regression) or mode (classification) as baseline

Feature engineering:

• Mean, median, mode can be features in aggregations

Population vs Sample Notation

Population mean: \mu = \frac{1}{N}\sum_{i=1}^{N} x_i

Sample mean: \bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i

The sample mean \bar{x} estimates the population mean \mu.

Key property: E[\bar{X}] = \mu (unbiased estimator)

Computational Considerations

Numerical stability for mean:

For very large datasets, use running mean:

\bar{x}_n = \bar{x}_{n-1} + \frac{x_n - \bar{x}_{n-1}}{n}

Efficiency for median:

• Full sort: O(n \log n)
• Selection algorithm: O(n) average case

Mode:

• Using hash map: O(n) time, O(k) space where k is unique values

Examples

Hints

1. Use np.mean and np.median for the first two values.
2. Count values with Counter, find the highest frequency, then take the smallest matching key.

Requirements

• Compute the arithmetic mean and median
• Choose the smallest value when the mode is tied
• Return exactly mean, median, and mode in a dictionary of Python floats

Constraints

• x is a nonempty one-dimensional numeric list
• Use NumPy and the Python standard library only

Starter Code

from collections import Counter
import numpy as np

def mean_median_mode(x: list) -> dict:
    """
    Returns a dictionary with mean, median, and mode.
    """
    # Write code here
    pass

Test Cases