Standard deviation is a statistic that measures the dispersion of a dataset relative to its mean and is calculated as the square root of the variance. The larger the data spread, the higher the standard deviation, indicating that data points are farther from the mean.
Key Takeaways
- Standard deviation measures the dispersion of a dataset relative to its mean.
- It is calculated as the square root of the variance.
- In finance, standard deviation is used as a measure of an asset’s riskiness.
- A volatile stock has a high standard deviation, while the deviation of a stable blue-chip stock is usually low.
- Standard deviation includes all uncertainty as risk, even favorable deviations such as above-average returns.
What Standard Deviation Measures
In finance, standard deviation sheds light on an investment’s historical volatility when applied to the annual rate of return of an investment. A higher standard deviation indicates a greater variance between each price and the mean, showcasing a larger price range.
Standard Deviation Formula
Standard deviation is calculated using the formula:
\begin{aligned}
&\text{Standard Deviation} = \sqrt{ \frac{\sum_{i=1}^n \left(x_i - \overline{x}\right)^2}{n-1} }\
&\textbf{where:}\
&x_i = \text{Value of the } i^{th} \text{ point}
&\overline{x} = \text{The mean value}
&n = \text{Number of data points}
\end{aligned}
Steps to Calculate Standard Deviation
- Calculate the mean of the data points.
- Determine the variance for each data point by subtracting the mean from each value.
- Square each result.
- Sum the squared values.
- Divide by the number of data points minus 1.
- Take the square root of this quotient.
Why Standard Deviation is Key in Risk Assessment
Standard deviation is critical in investing and trading strategies, helping measure market and security volatility. For example, an index fund likely has a low standard deviation versus its benchmark index, while aggressive growth funds may have a high standard deviation due to their higher risk and reward strategy.
Investors need to consider their tolerance for volatility and their overall investment goals when evaluating standard deviation. More aggressive investors might prefer higher volatility, while conservative investors may opt for lower volatility strategies.
Standard Deviation vs. Variance
Variance measures the spread between data points around the mean, with higher variance indicating larger gaps between values. Standard deviation simplifies this by taking the square root of the variance, making it easier to interpret and compare due to its common unit of measurement with the data.
Strengths of Standard Deviation
- Widely recognized and easily understood.
- Includes all data points, giving a complete analysis.
- Useful in further algebraic computations.
Limitations of Standard Deviation
- Does not measure the actual distance from the mean, only squared differences.
- Outliers can disproportionately affect results.
- Complex calculations increase the risk of computational errors.
Example of Standard Deviation
Imagine data points of 5, 7, 3, and 7, with a mean of 5.5. The variance after squaring each difference from the mean and summing these squares, divided by 3 (N-1), is approximately 3.67. The square root of this variance gives a standard deviation of about 1.915.
Interpretation of High Standard Deviation
A high standard deviation signals that data is widely spread around the mean, whereas a low standard deviation suggests data points are clustered closely around the mean.
Visualizing Data Distribution
Visually, data with a ‘fat’ distribution indicates a higher standard deviation compared to a ‘skinny’ distribution. Software tools like Excel simplify standard deviation calculations with built-in functions tailored for different data sets.
Bottom Line
Standard deviation helps investors gauge risk by understanding past volatility of returns, thus making informed decisions and recognizing underlying risks. Acknowledging the limitations and strengths of standard deviation is crucial for based on the complexity of their portfolios.
Related Terms: Variance, Mean, Historical Volatility, Index Fund, Benchmark, Aggressive Growth Funds.
References
- Netcials. “Apple Inc (AAPL) Stock 5 Years History”.
