How Does Excel Calculate Skewness

Calculate statistical skewness just like Microsoft Excel using the population or sample formula

Understanding how Excel calculates skewness is crucial for data analysts, statisticians, and business professionals who need to interpret the asymmetry of data distributions. This comprehensive guide explains the mathematical foundations, Excel’s specific implementation, and practical applications of skewness measurements.

What is Skewness?

Skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. It provides insight into the shape of data distribution:

• Positive skewness: The right tail is longer; the mass of the distribution is concentrated on the left
• Negative skewness: The left tail is longer; the mass of the distribution is concentrated on the right
• Zero skewness: The distribution is perfectly symmetrical around the mean

Excel’s Skewness Functions

Excel provides two distinct functions for calculating skewness, each serving different statistical purposes:

1. SKEW Function (Sample Skewness)

The SKEW function calculates the skewness of a sample dataset using the following formula:

SKEW = [n/(n-1)(n-2)] * Σ[(xᵢ – x̄)/s]³

Where:

• n = number of data points
• xᵢ = individual data points
• x̄ = sample mean
• s = sample standard deviation

2. SKEW.P Function (Population Skewness)

The SKEW.P function calculates the skewness of an entire population using this formula:

SKEW.P = (1/n) * Σ[(xᵢ – μ)/σ]³

Where:

• n = number of data points
• xᵢ = individual data points
• μ = population mean
• σ = population standard deviation

Key Differences Between SKEW and SKEW.P

Feature	SKEW (Sample)	SKEW.P (Population)
Statistical Context	Sample data (subset of population)	Complete population data
Denominator Adjustment	Uses n-1 in standard deviation	Uses n in standard deviation
Bias Correction	Includes [n/(n-1)(n-2)] factor	No bias correction factor
Typical Use Case	Inferential statistics	Descriptive statistics
Excel Version Introduced	Excel 2010	Excel 2013

Step-by-Step Calculation Process

Let’s examine how Excel computes skewness using a practical example with the dataset: 3, 5, 7, 8, 9

1. Calculate the Mean

First, Excel calculates the arithmetic mean (average) of the data points:

Mean (x̄) = (3 + 5 + 7 + 8 + 9) / 5 = 32 / 5 = 6.4

2. Compute Deviations from Mean

Next, Excel calculates how much each data point deviates from the mean:

Data Point (xᵢ)	Deviation (xᵢ – x̄)
3	3 – 6.4 = -3.4
5	5 – 6.4 = -1.4
7	7 – 6.4 = 0.6
8	8 – 6.4 = 1.6
9	9 – 6.4 = 2.6

3. Calculate Standard Deviation

For sample skewness (SKEW), Excel uses the sample standard deviation:

s = √[Σ(xᵢ – x̄)² / (n-1)] = √[(-3.4)² + (-1.4)² + (0.6)² + (1.6)² + (2.6)² / 4] ≈ 2.41

4. Compute Cubed Z-Scores

Excel then calculates the cubed standardized values (z-scores):

Deviation	Standardized (z)	Cubed (z³)
-3.4	-3.4/2.41 ≈ -1.41	(-1.41)³ ≈ -2.80
-1.4	-1.4/2.41 ≈ -0.58	(-0.58)³ ≈ -0.19
0.6	0.6/2.41 ≈ 0.25	(0.25)³ ≈ 0.02
1.6	1.6/2.41 ≈ 0.66	(0.66)³ ≈ 0.29
2.6	2.6/2.41 ≈ 1.08	(1.08)³ ≈ 1.26

5. Apply the Skewness Formula

For sample skewness (SKEW):

SKEW = [5/(5-1)(5-2)] * (-2.80 – 0.19 + 0.02 + 0.29 + 1.26) ≈ 0.42

For population skewness (SKEW.P):

SKEW.P = (1/5) * (-2.80 – 0.19 + 0.02 + 0.29 + 1.26) ≈ -0.28

Interpreting Skewness Values

The skewness value provides important information about your data distribution:

Skewness Range	Interpretation	Distribution Shape
Less than -1	Highly negative skew	Long left tail
-1 to -0.5	Moderate negative skew	Left tail present
-0.5 to 0.5	Approximately symmetric	Normal distribution
0.5 to 1	Moderate positive skew	Right tail present
Greater than 1	Highly positive skew	Long right tail

Practical Applications of Skewness

Understanding skewness has numerous practical applications across various fields:

1. Finance: Analyzing return distributions of assets to assess risk. Positive skewness indicates potential for extreme positive returns, while negative skewness suggests risk of extreme losses.
2. Quality Control: Monitoring manufacturing processes where symmetric distributions indicate consistent product quality.
3. Medical Research: Analyzing biological measurements where skewness can indicate health risks or treatment effects.
4. Market Research: Understanding customer behavior distributions to tailor marketing strategies.
5. Economics: Examining income distributions where positive skewness often indicates wealth inequality.

Common Mistakes When Calculating Skewness

Avoid these frequent errors when working with skewness calculations:

• Confusing sample and population functions: Using SKEW when you should use SKEW.P (or vice versa) can lead to incorrect interpretations.
• Ignoring data quality: Outliers can dramatically affect skewness calculations. Always clean your data first.
• Small sample sizes: Skewness measurements become unreliable with fewer than 30 data points.
• Misinterpreting zero skewness: A skewness of zero doesn’t necessarily mean a normal distribution, just symmetry.
• Overlooking units: Skewness is unitless, but the interpretation depends on the context of your data.

Advanced Considerations

1. Fisher vs. Pearson Skewness

Excel uses Fisher’s definition of skewness (the third standardized moment), but it’s important to understand Pearson’s alternative measures:

• Pearson Mode Skewness: (Mean – Mode) / Standard Deviation
• Pearson Median Skewness: 3*(Mean – Median) / Standard Deviation

2. Relationship with Kurtosis

Skewness and kurtosis together provide a complete picture of a distribution’s shape:

• Skewness measures asymmetry
• Kurtosis measures “tailedness” (heaviness of tails relative to normal distribution)

3. Transformations for Normality

When dealing with skewed data, these transformations can help normalize distributions:

Skewness Direction	Recommended Transformation	Example
Positive (right) skew	Logarithm, Square root	LOG(x), SQRT(x)
Negative (left) skew	Square, Cube	x², x³

Authoritative Resources on Skewness

For more in-depth information about skewness calculations and statistical distributions:

• NIST Engineering Statistics Handbook – Measures of Shape: Comprehensive government resource on statistical measures including skewness and kurtosis.
• Brigham Young University – Describing Distributions Numerically: Academic explanation of skewness with practical examples.
• CDC Glossary of Statistical Terms: Government health organization’s definitions of statistical concepts including skewness.

Excel Skewness Functions in Practice

Let’s examine how to use Excel’s skewness functions with real-world examples:

Example 1: Stock Market Returns

Consider monthly returns for a stock over 24 months: 1.2%, -0.5%, 2.1%, 0.8%, -1.5%, 3.2%, 0.5%, 1.8%, -0.3%, 2.5%, 0.9%, -1.1%, 4.0%, 1.2%, 0.7%, -0.8%, 1.5%, 2.3%, 0.6%, -0.2%, 1.9%, 3.1%, 0.4%, -1.3%

Using =SKEW(A1:A24) might return approximately 0.85, indicating moderate positive skewness – suggesting that while most returns are modest, there are occasional large positive returns.

Example 2: Household Income Data

For a sample of 50 household incomes (in thousands): [35, 42, 28, 55, 31, 48, 39, 62, 45, 33, 51, 40, 37, 58, 44, 30, 65, 36, 49, 53, 29, 47, 34, 56, 41, 32, 50, 60, 38, 46, 52, 27, 43, 59, 31, 54, 40, 35, 61, 57, 42, 39, 36, 48, 50, 33, 45, 55, 37, 41]

Applying =SKEW.P(B1:B50) might yield about 0.42, showing slight positive skewness typical of income distributions where most people earn moderate incomes but a few earn significantly more.

Limitations of Excel’s Skewness Functions

While Excel’s skewness functions are powerful, they have some limitations:

• No confidence intervals: Excel doesn’t provide confidence intervals for skewness estimates.
• Limited data handling: The functions can’t directly handle grouped data or frequency distributions.
• No visualization: You need to create separate charts to visualize the skewness.
• Precision limits: Excel uses 15-digit precision which may affect very large datasets.
• No bias correction options: Advanced statistical software offers more bias correction options.

Alternative Methods for Calculating Skewness

For more advanced analysis, consider these alternatives:

1. Using Data Analysis Toolpak

Excel’s Analysis ToolPak provides descriptive statistics including skewness:

1. Go to Data > Data Analysis
2. Select “Descriptive Statistics”
3. Choose your input range and check “Summary statistics”
4. The output includes both skewness and kurtosis

2. Manual Calculation

For complete control, you can implement the skewness formula manually:

1. Calculate the mean using =AVERAGE()
2. Calculate each deviation from the mean
3. Cube each standardized deviation
4. Sum the cubed values
5. Apply the appropriate formula factor

3. Using Other Software

Specialized statistical software often provides more robust skewness analysis:

• R: skewness() function in the moments package
• Python: scipy.stats.skew() function
• SPSS: Analyze > Descriptive Statistics > Descriptives
• Minitab: Stat > Basic Statistics > Display Descriptive Statistics

Conclusion

Understanding how Excel calculates skewness empowers you to make better data-driven decisions. Whether you’re analyzing financial returns, quality control metrics, or scientific measurements, skewness provides critical insights into the asymmetry of your data distribution. Remember that:

• Use SKEW for sample data when making inferences about a larger population
• Use SKEW.P when you have complete population data
• Always consider skewness in conjunction with other statistical measures
• Visualize your data to better understand the skewness
• Be aware of the limitations and potential biases in skewness measurements

By mastering these concepts and Excel’s implementation, you’ll be better equipped to interpret your data accurately and make more informed decisions based on the true shape of your distributions.