Average Calculator — Mean, Median & Mode

Formula

Mean = Sum / Count

Mean is the sum of all values divided by the count. Median is the middle value. Mode is the most frequent. Standard deviation measures spread.

Worked Examples

Example 1: Class Test Scores

Problem:Find mean, median, mode of: 85, 92, 78, 85, 90, 88, 85.

Solution:Mean = (85+92+78+85+90+88+85)/7 = 603/7 = 86.14 Sorted: 78, 85, 85, 85, 88, 90, 92 Median = 85 (middle value) Mode = 85 (appears 3 times)

Result:Mean: 86.14, Median: 85, Mode: 85

Example 2: Investment Returns

Problem:Calculate average return: 12%, -5%, 8%, 15%, 3%.

Solution:Sum = 12 + (-5) + 8 + 15 + 3 = 33 Mean = 33/5 = 6.6% Sorted: -5, 3, 8, 12, 15 Median = 8% Range = 15 - (-5) = 20%

Result:Average return: 6.6%, Median: 8%

Example 3: Standard Deviation

Problem:Find SD of: 2, 4, 4, 4, 5, 5, 7, 9.

Solution:Mean = 40/8 = 5 Variance = [(2-5)² + (4-5)² + ... + (9-5)²]/8 = [9+1+1+1+0+0+4+16]/8 = 32/8 = 4 SD = √4 = 2

Result:Standard Deviation = 2

Frequently Asked Questions

What is the difference between mean, median, and mode?

Mean is the sum divided by count (arithmetic average). Median is the middle value when sorted. Mode is the most frequent value. Each measures 'center' differently.

When should I use median instead of mean?

Use median when data has outliers or is skewed. Example: incomes (a few billionaires skew the mean). Median better represents 'typical' value in skewed distributions.

What is standard deviation?

Standard deviation measures spread from the mean. Low SD means data clusters near average; high SD means data is spread out. About 68% of data falls within 1 SD of mean.

What does 'no mode' mean?

If all values appear the same number of times (once each), there's no mode. Data can also have multiple modes (bimodal, multimodal) if several values tie for most frequent.

How do I calculate weighted average?

Multiply each value by its weight, sum the products, divide by sum of weights. Example: 80 (weight 2) and 90 (weight 3): (80×2 + 90×3)/(2+3) = 86.

What is variance?

Variance is the average of squared differences from the mean. It measures spread. Standard deviation is the square root of variance, in original units.

How do outliers affect averages?

Outliers heavily affect mean and standard deviation but barely affect median. One extreme value can make mean unrepresentative. Always check for outliers.

Can median be a value not in the dataset?

Yes, when there's an even number of values. The median is the average of the two middle values, which may not be in the original data.

What is the geometric mean?

Geometric mean = nth root of the product of n values. Used for growth rates, percentages. For 2 and 8: √(2×8) = 4. Always ≤ arithmetic mean.

Background & Theory

Averages measure the "center" of a dataset. Different types of averages are appropriate for different situations. **Measures of Central Tendency:** **Arithmetic Mean:** Mean = Sum of all values / Number of values Most common average, sensitive to outliers. **Median:** Middle value when data is sorted. Resistant to outliers. **Mode:** Most frequently occurring value. Only average that works with non-numeric data. **Measures of Spread:** **Range:** Max - Min Simple but sensitive to outliers. **Variance:** Average squared deviation from mean. σ² = Σ(x - μ)² / n **Standard Deviation:** √Variance Same units as original data. **When to Use Each:** | Situation | Use | |-----------|-----| | Normal data | Mean | | Skewed data | Median | | Categorical | Mode | | Comparing spread | Std Dev | **Key Formulas:** - Mean: x̄ = Σx / n - Variance: σ² = Σ(x - x̄)² / n - Std Dev: σ = √variance

History

The concept of average dates to ancient times. Babylonian astronomers averaged observations to reduce errors as early as 300 BCE. The arithmetic mean became formalized in the 17th century. The word "average" comes from Arabic "awar" (damaged goods), relating to maritime insurance where losses were averaged among merchants. Karl Pearson and Francis Galton developed modern statistics in the late 19th century, introducing standard deviation (1893) and regression to the mean. Ronald Fisher further developed statistical methods in the early 20th century. Today, averages are fundamental to science, business, sports, education, and daily life. From batting averages to GDP per capita, from grade point averages to climate data, measures of central tendency help us understand and compare data.

Average Calculator — Mean, Median & Mode | NovaCalculator