What is Standard Deviation?
Standard deviation quantifies the amount of variation in a set of data points. In other words, it tells us how much the individual data points deviate from the average value. A smaller standard deviation signifies that data points are closely packed together, while a larger one indicates a more spread-out dataset. Therefore the standard deviation tells you how spread out your data is from your mean.
Population vs. Sample
Before we dive deeper into standard deviation, let’s clarify an important distinction: population and sample.
A refers to the entire data set you’re interested in, while a “sample” is a subset of that population. When calculating the standard deviation, the formulas vary slightly depending on whether you’re working with population data or sample data. Don’t worry; we’ll cover how to calculate the standard deviation for both scenarios later.
Real-world Example. – Standard Deviation
Imagine we’re comparing the average height of two groups of basketball players from different schools, School A and School B. We found that both schools have an average height of 6 feet, but we want to understand whether the data is similar between both groups.
Here’s where standard deviation comes in. School A has a standard deviation of 2 inches, while School B has a standard deviation of 6 inches. From this, we can infer that School A’s basketball players’ heights cluster more closely around 6 feet, while School B’s players have a more varied height range.
This could be the same for 3-point scoring averages. If the standard deviation is higher, the team may score 2% in one game and 40% in the next game. A high standard deviation shows we are missing consistency and one of the key issues when you find your have a large standard deviation.
The Standard Deviation Formula
Standard deviation can be calculated manually or through Excel (STDDEV function) or other statistical tools. You can also download our Standard Deviation Calculator, part of our Lean Six Sigma Yellow Belt Course.
The Standard Deviation Formulas
For Population Data (for entire population)
Standard Deviation (σ) = √Σ(x – μ)² / N
Where:
– σ is the population standard deviation.
– Σ is the sum of all elements.
– x is an individual data point.
– μ is the population mean (average).
– N is the number of data points.
For Sample Data
Standard Deviation (s) = √Σ(x – x̄)² / (n – 1)
Where:
– s is the sample standard deviation.
– Σ is the sum of all elements.
– x is an individual data point.
– x̄ is the sample mean (average).
– n is the number of data points.
Calculating standard deviation
The standard deviation is calculated as follows:
Calculate the mean of all data points. The mean is calculated by adding all the data points and dividing them by the number of data points.
Calculate the variance for each data point. The variance for each data point is calculated by subtracting the mean from the value of the data point.
Square the variance of each data point (from Step 2).
Sum of squared variance values (from Step 3).
Divide the sum of squared variance values (from Step 4) by the number of data points in the data set less 1.
Take the square root of the quotient (from Step 5).
Tip: When you calculate the difference between the data values and the mean, you often get a negative number. To remove the negative number when we calculate standard deviation, we use the square functionary (squared deviations) to remove the negative figures. However, to keep the standard deviation within the same units, we must then apply the square root of the variance.
Don’t get confused. We square the data values to turn the negatives into positives but to end up with the same unit size, we then square root the result to get the final standard deviation.
Why Do We Use n-1 in the Standard Deviation Calculation?
When calculating the standard deviation for a sample of a population, we use n-1 instead of n in the denominator of the formula. This practice is referred to as Bessel’s Correction. But why do we do this?
Simply put, when we take a sample, we’re trying to make an estimation about a larger population. However, a sample might not perfectly represent the population’s true characteristics. By using n-1, we’re essentially adjusting our estimate to account for the fact that there’s an increased likelihood of error when dealing with samples.
This adjustment is especially important when we’re working with small sample sizes. If we were to use n instead of n-1, it could lead to an underestimation of the variability in the data. Thus, Bessel’s correction, or using n-1, provides a better estimate of the true standard deviation of the population when calculated from sample data.
In conclusion, the use of n-1 is a corrective measure that helps to ensure the accuracy of our standard deviation calculation when working with sample data. It helps in providing a less biased and more reliable estimate of the variability inherent in the larger population.
What Does Standard Deviation Tell You?
Standard deviations describe the degree of dispersion of data in an area. The standard deviation returns a calculated amount to describe where the information is found. Normal distributions indicate the deviation between values from the mean.
What Does a High Standard Deviation Mean?
What is a large standard deviation indicates is that the data points are spread out over a wider range, away from the mean, and thus, the data is more dispersed. This could suggest a larger variety or volatility in the data set. For example, in the stock market, a high standard deviation of a stock’s price would mean that the price is highly volatile, leading to a higher risk and potential for significant returns. Contrastingly, a low standard deviation suggests that data points are closer to the mean, indicating less variability or volatility. In our previous basketball example, a team with a high standard deviation in players’ heights would have more diverse heights amongst its players. Understanding the implication of a high standard deviation allows for a more nuanced interpretation of data analysis results.
What Does a Low Standard Deviation Mean?
A low standard deviation indicates that the data points are tightly clustered around the mean, suggesting less dispersion or variability in the dataset. This implies relative consistency or uniformity in the data set. For instance, in a production line, a low standard deviation in the size of produced items signifies high consistency and quality control. Similarly, a mutual fund with a low standard deviation of returns would indicate less risk as the returns are relatively stable, not deviating much from the average return. In the context of our basketball example, a team with a low standard deviation in players’ heights would mean the heights are relatively similar, possibly indicating a specific recruitment strategy. Comprehending the meaning of a low standard deviation is crucial in gaining deeper insights from data.
Is a Standard Deviation of 1 Good or Bad?
The question of whether a standard deviation of 1 is good or bad is largely context-dependent and depends immensely on the type of data you’re dealing with. In a normal distribution, a standard deviation of 1 means that approximately 68% of data points fall within one standard deviation of the mean (average), while about 95% lie within two standard deviations, and roughly 99.7% are within three standard deviations.
This can be beneficial when you want to know how much variance there is in a dataset. If there’s a low standard deviation (close to 1 or lower), it suggests that the data points tend to be closer to the mean, indicating low variance. This might be considered “good” in contexts where consistency or predictability is desired.
Conversely, a high standard deviation (significantly higher than 1) indicates that data points spread out over a wider range, signifying high variability. This could be “bad” in situations where you want low variance but “good” when you are looking for a high degree of diversity or dispersion in your data.
In the end, the goodness or badness of a standard deviation of 1 is largely subjective and must be interpreted based on the specific context, the type of data you’re dealing with, and what you’re hoping to understand from this data.
Strengths of Standard Deviation
Standard deviation, as a statistical tool, carries several strengths that make it a popular choice for data analysts and researchers alike.
Unveils Data Spread
Firstly, standard deviation provides deep insights into the spread of data. It offers a numerical measure of how dispersed or concentrated the data points are around the mean. This is invaluable in scenarios where understanding variability matters just as much as, if not more than, the average.
Facilitates Comparisons
Secondly, standard deviation facilitates comparisons between different sets of data, even if they have the same mean. As seen in the basketball player example above, it helps distinguish between data sets where the mean might be identical but the spread of data points is different.
Basis for Other Statistical Measures
Thirdly, standard deviation is the basis for other important statistical measures and concepts. For instance, the concept of Z-score or standard score, which measures the number of standard deviations a data point is from the mean, is based on standard deviation. Similarly, standard deviation plays a key role in the calculation of confidence intervals and hypothesis testing.
Useful in Normal Distributions
Lastly, standard deviation has a particular significance in a normal distribution (a symmetrical bell-shaped distribution of data). Approximately 68% of data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. This property makes standard deviation a powerful tool for understanding and interpreting data in a normal distribution. The bell curve follows the three sigma rule with 3 sigma below and 3 sigma above the mean.
In conclusion, the strengths of standard deviation are manifold. Its ability to quantify data spread, facilitate comparisons, form the basis for other statistical measures, and its particular significance in normal distributions, make it a cornerstone of statistical analysis. Remember, that the standard deviation assumes a normal curve.
Limitations of Standard Deviation
Despite its many strengths, the standard deviation is not without its limitations, and it’s essential to be cognizant of these when using it as a statistical tool.
Sensitive to Outliers
Firstly, the standard deviation is highly sensitive to outliers in the data. Even a single outlier can significantly inflate the standard deviation, providing a distorted picture of the dispersion in the data. Therefore, in data sets with extreme values or outliers, using standard deviation as a measure of spread may not be accurate.
Assumes a Normal Distribution
Secondly, standard deviation assumes a normal distribution of data. This means the unique properties of standard deviation (like the rule that 68% of data falls within one standard deviation of the mean) only strictly hold in normally distributed data sets. If the data is not normally distributed, the interpretation of standard deviation becomes more complex.
Inability to Differentiate Between Data Sets
Thirdly, the standard deviation cannot differentiate between two data sets with the same spread but different patterns. For example, a data set with values clustered closely around the mean and another with values spread uniformly across the range can have the same standard deviation despite their different distributions.
Not a Standalone Measure
Lastly, the standard deviation is not a standalone measure. It is most effectively combined with other statistical measures like mean and median. Used alone, the standard deviation can often fail to provide a complete picture of the data.
While the standard deviation is a powerful statistical tool, it’s crucial to be aware of its limitations and use it judiciously in combination with other statistical measures to accurately understand your data.
Uniform Distribution
In statistics, the uniform distribution represents equally distributed data across a range. All values within this given range have the same probability of occurrence. When dealing with uniform distribution, the standard deviation calculated can be low or high depending on how spread the distribution is.
The main takeaway is that the standard deviation of a uniformly distributed dataset provides insights into the overall variability of the dataset.
Tips & Tricks for Interpreting Standard Deviation
Interpreting standard deviation can be complex, but here are some tips and tricks that can help you gain a more nuanced understanding.
Understand the Context: Always interpret standard deviation in the context of your data. A high standard deviation could mean high variability, while a low standard deviation might suggest consistency. Depending on what you’re studying, either could be ‘good’ or ‘bad’.
Examine the Distribution: Standard deviation is most meaningful when applied to data that is normally distributed. If your data doesn’t follow a normal distribution, consider other statistical measures of spread, such as the interquartile range.
Be Cautious with Outliers: Since standard deviation is sensitive to outliers, a single extreme value can drastically affect it. Always check your dataset for outliers before making conclusions based on standard deviation.
Use in Conjunction with Other Statistics: Standard deviation is most informative when used alongside other statistics, such as the mean or median. This will provide you with a fuller picture of your data.
Check for Uniform Distribution: If your data is uniformly distributed, remember that the standard deviation can be high or low, depending on the range of the distribution. The key is to understand what the standard deviation is signifying about the spread of your data in this context.
Remember, the key to interpreting standard deviation is context. It’s not about the raw number, but what that number tells you about your data
