How They Differ and Practical Uses in Finance and Investing
Standard Error of the Mean vs. Standard Deviation
Suppose you’re choosing between two jobs. You’re told both pay an average of $5,000 a month, but there’s a catch: Job A has a traditional salary that pays $5,000 every month according to a contract. Job B is gig work, where you might earn $7,500 one month and $2,000 the next. They have the same average but mean something very different when you are planning your rent or mortgage payments.
Situations like this are why statistical measures like standard deviation (often symbolized as σ) and standard error of the mean (SEM) are employed—they give you more depth than simple averages. Standard deviation tells you how wild those income swings might be. In our example, Job A’s steady salary would have a smaller standard deviation (in fact, none at all month to month), while Job B’s unpredictable gig income would have a large one.
SEM tackles a different question: how much can you trust the figure of $5,000 per month? If Job A’s average comes from tracking hundreds of employees over several years, but Job B’s is based on just a few gig workers’ experiences last month, that’s crucial information for your decision. Most succinctly put, standard deviation is about where the data are clustered in one sample data, while SEM is where the means would be clustered around many samples taken of a given set of things. We tackle both below.
Key Takeaways
- Standard deviation describes how much variability—or fluctuation—exists within a data set.
- The standard error of the mean (SEM) indicates how accurately a data set represents the true population by comparing the dataset’s average to the population’s average.
- The size of the data set—the sample size—doesn’t affect standard deviation, but the sample size is a key factor in calculating the SEM.
- A larger random sample will always provide a mean that is closer to the mean of the population.
How SEM and Standard Deviation Are Used
Both standard deviation and SEM are widely used in scientific research, business, and finance to measure the variability and estimate the reliability of data. They are usually reported together.
Using Standard Deviation
Standard deviation is employed in everything from sports analytics to finance and investing. In finance, standard deviation is key to helping businesses and investors quantify risk and, therefore, the return they would require from an investment to make it worthwhile.
In modern portfolio theory, standard deviation is used to determine the range of possible outcomes of future performance, both for individual assets and portfolios as a whole. Another way of putting it is that it measures volatility, which is a way of measuring risk. The higher the standard deviation of possible outcomes or volatility, the greater the risk.
Let’s return to our example of the jobs available. Say you’re looking at that gig work position where your monthly income bounces around. The company tells you the average (mean) monthly income is $5,000, and the standard deviation is $1,000. But what does that actually mean for your bank account?
The standard deviation of $1,000 tells you that about two-thirds of your monthly paychecks will fall within $4,000 to $6,000 ($5,000 ± $1,000). But some months will fall outside this range. You might have a great payday one month at $6,100, but you must also be prepared for those tougher months when you make $4,100.
This is practical knowledge: With a steady job, you might only need to keep one month’s expenses as a safety net. That’s because its standard deviation should be zero, set by contract. But with your gig work and its $1,000 standard deviation, you’d need a bigger emergency fund to handle those low-income months.
Using Standard Error of the Mean
SEM is commonly used in scientific studies, medical studies, and clinical trials, as well as political surveys and numerous other fields. In finance and investing, SEM is used to judge the consistency—or uncertainty—of estimates such as average returns, risk, and economic indicators by assessing the reliability of the sample data set.
For example, investors or fund managers often use historical data to calculate projected average returns of a stock or index. They use SEM to calculate how much those future returns are likely to fluctuate based on the data sample used, including different time frames, market conditions, etc.
While standard deviation reveals how much variation exists within a single sample, SEM shows how likely the mean of the sample is correct. The smaller the SEM, the closer your sample’s mean is to the actual average of the whole population (remembering that in statistics, “population” isn’t just people but might be all monthly salaries for a specific job).
Here’s the formula:
SEM=nσ
Where:
- σ represents the standard deviation of the population. In practice, we rarely know the true population standard deviation. We usually use the standard deviation of our sample as an estimate.
- n is the sample size—the number of data points in your sample.
- √n is the square root of the sample size.
While online calculators can take care of the above for you, the concept is straightforward: the smaller the sample size, the less chance you’re matching the real population.
Coming back to our job example, let’s say you’re not just considering one gig work opportunity, but you’re researching the income of all gig workers in your field. You can’t possibly survey everyone, so you take a sample. Let’s say you survey 100 gig workers and find their average (mean) monthly income is $5,000, with an SEM of $100.
The SEM of $100 is not about your individual salary but about how reliable that $5,000 average is for all gig workers. The SEM of $100 tells you that if you were to repeat your survey many times, taking different samples of 100 gig workers each time, the average of those sample means would cluster around the true population mean. Specifically, about 68% of those sample means would fall within $4,900 and $5,100 ($5,000 ± $100). About 95% of them would fall within two standard errors ($4,800 to $5,200, or $5,000 +/- $200.
This is practical knowledge when examining research. Imagine another researcher surveys only 25 gig workers and also reports an average income of $5,000. Because their sample size is smaller, their SEM will likely be larger (let’s say, $200). This means their $5,000 average is less reliable than yours. There’s a wider range of uncertainty around their estimate of the true population mean.
The SEM gives you a sense of how much confidence you can place in the reported average. A smaller SEM means more confidence, indicating that the sample mean is a good representation of the actual population mean. A larger SEM indicates greater uncertainty, meaning the true population mean could be very different from the sample’s average.
The Bottom Line
Both standard deviation and standard error of the mean are commonly used measures of variability in finance and investment as well as any field that requires statistical analysis. In investing, both are used to assess risk: standard deviation looks at how significant the fluctuations within a data set are, and SEM assesses how reliable the mean is compared with the actual population.