Mean, Median, Mode

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Mean (Arithmetic Mean)

Definition

The arithmetic average of all values, representing the central tendency of the data.

Formula

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

Example

For the numbers: $2, 4, 6, 8$

\text{Mean} = \frac{2 + 4 + 6 + 8}{4} = \frac{20}{4} = 5

Key Points

Sensitive to extreme values (outliers)
Best used when data is symmetrically distributed

Comparison of Different Means

Try Geometric Mean Try Harmonic Mean

Important Relationship:

For any set of positive numbers:

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

(Equality occurs only when all numbers are the same)

Type	Formula	Example
Arithmetic Mean	$\frac{1}{n}\sum_{i=1}^n x_i = \frac{x_1 + x_2 + ... + x_n}{n}$	$\frac{2 + 4 + 6 + 8}{4} = 5$
Geometric Mean	$\sqrt[n]{\prod_{i=1}^n x_i} = \sqrt[n]{x_1 \times x_2 \times ... \times x_n}$	$\sqrt[4]{2 \times 4 \times 6 \times 8} \approx 4.4$
Harmonic Mean	$\frac{n}{\sum_{i=1}^n \frac{1}{x_i}} = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + ... + \frac{1}{x_n}}$	$\frac{4}{\frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8}} \approx 3.8$

Arithmetic Mean(Current)

Best For:

Simple averages of quantities
Calculating central tendency
Equal weight to all values
Linear relationships

Common Uses:

Test scores
Heights/weights
Daily temperatures
Income levels

Limitations:

Sensitive to outliers
Not ideal for ratios or rates

Geometric Mean

Best For:

Growth rates
Percentage changes
Ratios and proportions
Exponential relationships

Common Uses:

Investment returns
Population growth
Interest rates
Price indices

Limitations:

Only works with positive numbers
More complex calculation

Harmonic Mean

Best For:

Rates and speeds
Per-unit quantities
Reciprocal relationships
Density measurements

Common Uses:

Average speed over different distances
Price per unit calculations
Density measurements
Circuit calculations (parallel resistors)

Limitations:

Only works with positive numbers
Gives more weight to smaller values

When to Use Each Mean

Use Arithmetic Mean when you need a simple average and all values should have equal weight

Use Geometric Mean when dealing with growth rates, returns, or multiplicative changes

Use Harmonic Mean when working with rates, speeds, or other measures where using reciprocals makes sense

Median

Definition

The middle value in a sorted set of numbers. For an even number of values, it's the average of the two middle numbers.

Formula

\text{Median} = \begin{cases} x_{(n+1)/2} & \text{if n is odd} \\ \frac{x_{n/2} + x_{n/2 + 1}}{2} & \text{if n is even} \end{cases}

Example

For odd number of values: $1, 3, 5, 7, 9$

\text{Median} = x_{(n+1)/2} = x_{(5+1)/2} = x_3 = 5

For even number of values: $1, 3, 5, 7$

\text{Median} = \frac{x_{n/2} + x_{n/2 + 1}}{2} = \frac{x_{4/2} + x_{4/2 + 1}}{2} = \frac{x_2 + x_3}{2} = \frac{3 + 5}{2} = 4

Key Points

Not affected by extreme values (outliers)
Best used when data is skewed or contains outliers

Mode

Definition

The most frequently occurring value(s) in a set of numbers. A dataset can have no mode, one mode, or multiple modes.

Formula

\text{Mode} = \text{value(s) with the highest frequency}

Example

For the numbers with multiple modes: $1, 2, 2, 3, 4, 4, 5$

\text{Mode} = 2, 4

Because 2 and 4 each appear twice, more than any other number

For numbers $1, 2, 3, 4$ , there is no mode because all numbers appear exactly once.

Key Points

Can have multiple modes (bimodal, trimodal, etc.) or no mode
Particularly useful for categorical data and discrete numerical data

Calculating Mean, Median, and Mode from a Bar Chart

Mean Calculation

1. Multiply each value by its frequency:

2 × 1 = 2
4 × 2 = 8
6 × 3 = 18
8 × 2 = 16
10 × 1 = 10

2. Sum all products: 2 + 8 + 18 + 16 + 10 = 54

3. Sum all frequencies: 1 + 2 + 3 + 2 + 1 = 9

4. Divide: 54 ÷ 9 = 6

Mean = 6

Median Calculation

1. List all values (including duplicates):

2, 4, 4, 6, 6, 6, 8, 8, 10

2. Find middle position: (n+1) ÷ 2 = (9+1) ÷ 2 = 5th

3. Find 5th value in ordered list

Median = 6

Mode Calculation

1. Count frequency of each value:

2: occurs 1 time
4: occurs 2 times
6: occurs 3 times
8: occurs 2 times
10: occurs 1 time

2. Find highest frequency (3)

3. Identify value(s) with that frequency

Mode = 6

Final Results:

Mean = 6.00

Median = 6

Mode = 6

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Mean, Median, Mode

Calculator

1. Load Your Data

2. Select Columns & Options

Learn More

Mean (Arithmetic Mean)

Definition

Formula

Example

Key Points

Comparison of Different Means

Arithmetic Mean(Current)

Best For:

Common Uses:

Limitations:

Geometric Mean

Best For:

Common Uses:

Limitations:

Harmonic Mean

Best For:

Common Uses:

Limitations:

When to Use Each Mean

Median

Definition

Formula

Example

Key Points

Mode

Definition

Formula

Example

Key Points

Calculating Mean, Median, and Mode from a Bar Chart

Mean Calculation

Median Calculation

Mode Calculation

Final Results:

Related Links

Range, Variance, Standard Deviation Calculator

Skewness Calculator

Histogram

Bar Chart

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