What Is Standard Deviation?

What Is Standard Deviation?

Let me explain standard deviation directly: it's a statistical measure that shows how much individual values in a dataset spread out from the average, or mean. You calculate it by taking the square root of the variance.

When data points are far from the mean, you see a higher deviation in the set, indicating more spread.

Key Takeaways

Standard deviation measures the dispersion of dataset values relative to the mean. You get it as the square root of the variance. In finance, it often gauges the relative riskiness of an asset—a volatile stock shows a high standard deviation, while a stable blue-chip stock has a low one. Businesses apply it to assess risk, manage operations, and plan cash flows based on seasonal changes and volatility.

Understanding Standard Deviation

Standard deviation is a key statistical tool I often see used in finance, especially for investing. When you apply it to an investment's annual rate of return, it reveals the historical volatility, showing how much the price has fluctuated over time.

A greater standard deviation means larger variance between each price and the mean, pointing to a wider price range. For instance, a volatile stock has a high standard deviation because its price swings up and down a lot, whereas a stable blue-chip stock has a low one, keeping its price steady.

You can also use standard deviation to predict performance trends. In investing, an index fund aims to match a benchmark index, so it should have a low standard deviation from that benchmark's value. Aggressive growth funds, however, often show a high standard deviation from stock indices because their managers take bold bets for higher returns. This higher deviation ties directly to the risk level investors face.

Analysts, portfolio managers, and advisors rely on standard deviation as a fundamental risk measure. Investment firms report it for their mutual funds and products. A large dispersion indicates how much the fund's return deviates from expected normals, and since it's straightforward, they share this stat with clients and investors.

Be aware, though: standard deviation treats all uncertainty as risk, even favorable ones like above-average returns.

Standard Deviation Formula

To calculate standard deviation, you take the square root of a value from comparing data points to the population's mean. The formula is: Standard Deviation = √[∑(xi - x̄)² / (n - 1)], where xi is the value of the ith point, x̄ is the mean, and n is the number of points.

Calculating Standard Deviation

Here's how you calculate it step by step. First, find the mean by adding all data points and dividing by their count. Then, for each point, subtract the mean to get the variance. Square those variances next. Sum the squared values, divide by n-1, and finally take the square root of that result.

Key Properties of Standard Deviation

One property is additivity, meaning you're comparing many data points for higher accuracy rather than single ones. Another is scale invariance, useful for comparing variability across datasets with different units, like inches versus centimeters, without conversions.

It also has symmetry and non-negativity: standard deviation is always positive and symmetrically distributed around the mean, balancing deviations above and below for overall dataset equilibrium.

Standard Deviation vs. Variance

Variance and standard deviation are linked—variance comes from averaging the squared differences from the mean, and standard deviation is its square root. Variance shows the spread size compared to the mean; larger variance means more variation and bigger gaps between values.

When values are close, variance is smaller, but it's harder to interpret since it's squared and not in the original units. Standard deviation is easier to visualize and apply because it's in the same units as the data.

Statisticians use it to check for normal distributions: in a normal curve, 68% of points fall within one standard deviation of the mean. Larger variances push more points outside, while smaller ones keep them close. Graphically, it's the width of the bell curve around the mean—wider curve means larger deviation.

How Standard Deviation Is Used in Business

Standard deviation goes beyond investing; businesses use it to assess risk, predict outcomes, and manage operations. In risk management, it quantifies volatility in operations, like measuring product return risks.

For financial analysis, it evaluates variability in performance metrics, such as investment return volatility, helping decide risk-return strategies and capital deployment.

In forecasting, it assesses sales data variability to spot seasonality and plan cash flows. For quality control, like in Six Sigma, it monitors process capability to reduce defects and improve manufacturing.

In project management, it evaluates performance, manages risks, and analyzes critical paths or earned value to track progress and quantify variances.

Strengths and Limitations of Standard Deviation

Standard deviation has strengths like being commonly used, so many professionals know it better than other deviation measures. It includes all data points, making it more robust than something like range, which ignores intermediates.

You can combine standard deviations from datasets using a specific formula, and it allows further algebraic uses, adding versatility. However, it doesn't directly measure how far points are from the mean—it uses squared differences instead.

Outliers heavily influence it because differences are squared, giving extremes more weight. Calculating it manually involves several steps, risking errors, though tools like Excel or Bloomberg can help. In Excel, use STDEV.S for numeric data or STDEVA for including text or logical values.

Examples of Standard Deviation

Take data points 5, 7, 3, and 7. Sum them to 22, divide by 4 for mean 5.5. Subtract mean from each: -0.5, 1.5, -2.5, 1.5. Square them: 0.25, 2.25, 6.25, 2.25. Sum to 11, divide by 3 for variance 3.67, square root about 1.915.

For Apple stock returns: 88.97% (2019), 82.31% (2020), 34.65% (2021), -26.41% (2022), 28.32% (April 2023). Mean 41.57%. Differences squared sum to 0.882, divide by 4 for variance 0.220, square root 0.4690 or 46.90%.

What Does a High Standard Deviation Mean?

A high standard deviation means big spread around the mean; low means data clusters tightly.

What Does Standard Deviation Tell You?

It shows data dispersion, comparing points to the mean and indicating if they're close or spread out. In normal distributions, it reveals how far values stray from the mean.

How Do You Find the Standard Deviation Quickly?

Look at the distribution graph: fatter shapes have larger deviations. Or use Excel's built-in functions for your dataset.

Is Lower Standard Deviation Better In Investing?

Not necessarily—lower means less risk, but investors' preferences vary by volatility tolerance and goals. Aggressive ones might prefer higher for potential returns.

The Bottom Line

Standard deviation assesses risk in business and investing by measuring data spread from the mean, indicating volatility. Investors use it for stability predictions, businesses for operations and planning. Consider its strengths and limitations when applying it.