Population standard deviation (pop SD) is a key statistical measure used in inferential statistics. It refers to the standard deviation of a population of data, describing how dispersed the values are from the mean. Whereas sample standard deviation (sample SD) measures the spread in a sample, population SD measures spread in the entire population.
Knowing the population SD is useful when you want to draw conclusions about a population from a sample. However, in practice, the true population SD is often not known. Population SD is usually estimated from the sample SD. Understanding the difference between sample SD and population SD is important for proper statistical analysis.
This article will provide a comprehensive look at population SD – its mathematical definition, how it compares to sample SD, its common uses and advantages, limitations, and how it relates to other statistical measures.
Origins
The symbol for population standard deviation (pop SD) has its origins in the early development of statistical theory and probability. The concepts of variance and standard deviation were first introduced in the 1920s and 1930s as ways to measure dispersion and variability in a data set.
The lower case Greek letter sigma squared (σ2) began to be used to denote population variance, while the lower case Greek letter sigma (σ) was used to represent the population standard deviation. This was in line with common mathematical conventions at the time of using Greek letters to denote population parameters.
The “pop” terminology to distinguish population versus sample values appears to have emerged later, in the 1950s and 1960s. As statistical sampling theory advanced, it became important to differentiate between variances and standard deviations calculated for an entire population versus those calculated from a sample of a population.
The “pop SD” symbol therefore combines the original Greek sigma notation for population standard deviation with the “pop” prefix used in modern statistics. It first came into widespread usage in the 1960s and 1970s and has since become the standard symbol for referring to population standard deviation.
As noted in this statistics text, using Greek letters for population parameters and Roman letters for sample statistics has become a broadly accepted convention to distinguish the two concepts.
Mathematical Definition
Pop SD stands for “population standard deviation”. Mathematically, it is the standard deviation of a probability distribution that represents an entire population. The pop SD is calculated using the following formula:
σ = √(∑(x – μ)2 / N)
Where:
- σ = Pop SD
- ∑ = Sum
- x = Individual data values
- μ = Population mean
- N = Total number of data values in the population
So in summary, to calculate the pop SD you first find the difference between each data value and the population mean. You square each of these differences, sum them all together, divide by the population size, and finally take the square root. This gives you the standard deviation for the entire population.
For example, say you had a population with 5 data values: 2, 4, 6, 8, 10. The mean would be 6. Plugging this into the formula:
σ = √((2-6)2 + (4-6)2 + (6-6)2 + (8-6)2 + (10-6)2) / 5)
= √(16 + 4 + 0 + 4 + 16) / 5
= √40 / 5
= 2
So the pop SD for this simple 5 value population would be 2. Understanding the mathematical calculation helps explain what pop SD means and how it quantifies dispersion in a population.
Compared to Sample SD
The population standard deviation (σ) differs from the sample standard deviation (s) primarily in that the population parameter is calculated from the entire population, while the sample statistic is calculated from a subset sample of the population. The formulas differ slightly, with the sample SD using (n-1) in the denominator rather than n for the population SD.
The sample SD tends to slightly overestimate the population SD, as it doesn’t have full information about the entire population. The sample SD also has greater variability from sample to sample. However, for large sample sizes (>30) the sample SD converges very closely to the true population SD.
Situations where the population SD is known are rare outside of simulated data sets. In practice, the sample SD is much more widely used as the population values are often not fully known. Still, the conceptual difference between the two measures is important to understand.
Overall, the sample SD acts as a good estimator of the population SD, especially at large sample sizes. The formulas differ only slightly, but represent important conceptual distinctions between population and sample-based parameters and statistics in statistics.
Usages
The population standard deviation (SD) is commonly used in statistical analysis to understand how spread out a dataset is. Some examples of when population SD is used include:
Calculating confidence intervals. Population SD allows you to calculate the margin of error around a sample mean to determine a range that likely contains the true population mean. This is useful in survey analysis and polling.
Determining statistical significance. Population SD is used in hypothesis testing to calculate a test statistic that indicates whether an effect is statistically significant or likely due to chance.
Making inferences about the population. The population SD gives information about the variability of data in the overall population. A low SD indicates data points tend to be clustered around the mean, while a high SD indicates the data is more spread out.
Standardizing data. Population SD can be used to standardize data values into z-scores. This allows comparing data from different distributions by expressing values in terms of standard deviations from the mean rather than raw scores.
Power analysis. Knowledge of the population SD helps determine the sample size needed to achieve a desired power for detecting a particular effect size in hypothesis testing.
Modeling data. In forecasting, predictive modeling, and simulations, assuming a population with a known SD allows generating realistic synthetic data.
Assessing normality. Comparing the sample SD to the assumed population SD helps determine if a sample matches the expected normal distribution.
Advantages
The population standard deviation (pop SD) has some notable advantages compared to other statistical measures of variability like sample standard deviation or variance.
First, pop SD provides an absolute and standardized measure of dispersion for an entire population dataset (cite: https://www.quora.com/Why-using-N-1-instead-of-N-when-calculating-sampling-standard-deviations-but-using-N-in-calculating-population-standard-deviation). Since it uses the full population, it does not depend on sample size or sampling variability like sample SD. This makes it more robust and reliable for describing the true dispersion in a population.
Additionally, pop SD is more intuitive and easier to interpret than variance, since it is expressed in the same units as the original data rather than squared units. This makes it easier for most people to understand the meaning of the pop SD value (cite: https://www.aacc.org/~/media/files/meetings-and-events/resources-from-past-events/conferences/2012/lab-automation/dec-6/hurdless_and_benefits_handout_dec_6_2012.pdf?la=en).
Overall, pop SD provides a robust, standardized measure of dispersion for an entire population dataset. It avoids sampling variability issues and is more intuitive to interpret than variance.
Limitations
While pop SD is a useful statistical measure, there are some limitations and potential downsides to relying solely on it:
Pop SD only shows the spread or variability in a population. It does not provide information about skewness or the shape of the distribution. For example, two populations can have the same pop SD but different skewness, so one variable may have more extreme outliers than the other. Relying only on pop SD would fail to reveal these differences.
Pop SD is most useful when the population data follows a normal distribution. For non-normal distributions, pop SD may not fully capture key features like heavy tails. Additional descriptive statistics like skewness and kurtosis may be needed.
Pop SD treats all deviations from the mean equally. But in some cases, larger deviations may be more important. Measures like Mean Absolute Deviation (MAD) can complement pop SD by emphasizing larger deviations.
Pop SD only looks at variation around the mean. But the location of the central tendency may also be important. Measures like quantiles and ranges are needed to fully describe the location and spread.
Sample SD is often used when population SD is unknown. But sample SD is a less precise estimate, especially for small samples. Over-reliance on sample SD can lead to incorrect conclusions.
Overall, while pop SD is an important basic measure of variability, relying solely on it can miss other key features of the distribution. Using pop SD in combination with other descriptive stats provides a more complete picture.
Common Confusions Around Population Standard Deviation
There are a few common misconceptions that people have around population standard deviation (pop SD):
Some mistakenly believe that pop SD can only be calculated if you have data on the entire population. However, pop SD is a parameter that describes the spread of values across an entire population. Even if you only have a sample of data, you can still estimate the pop SD using statistical methods. The key point is that pop SD is intended to represent the true standard deviation in the population, not just the sample.
Another misconception is thinking that pop SD is always larger than sample SD. In reality, the sample SD can be larger or smaller than the pop SD. Sample SD is susceptible to random variation based on the specific sample drawn, while pop SD reflects the true spread of values. For normally distributed data, the sample SD will on average converge to the pop SD as the sample size increases.[1]
Some also mistakenly use pop SD and sample SD interchangeably. However, there are important mathematical differences in how they are calculated, so they should not be treated as equivalent. Proper terminology is essential, especially in research and statistics.[2]
In summary, pop SD has a distinct definition from sample SD and the two should not be confused. Accurately understanding the terminology and mathematical foundations is key.
Related Statistical Measures
Population standard deviation is related to but differs from other statistical measures of spread and variation including sample standard deviation, variance, and range.
While population standard deviation (pop SD) measures the spread of values around the mean of an entire population, sample standard deviation measures the spread for a sample drawn from a population. Sample SD tends to slightly overestimate the variability in a population, while pop SD gives the exact standard deviation for the full population.
Population variance (pop var) is the square of pop SD, measuring the average of squared deviations around the mean. So while pop var also measures variability, taking the square root via pop SD makes the units more interpretable.
Range measures the spread between the minimum and maximum values in a dataset. While simple to calculate, range only looks at the outliers and doesn’t account for the overall distribution. Pop SD looks at all values relative to the mean, providing a more comprehensive measure of spread.
Understanding when to use pop SD versus related measures provides deeper insight into the variability and patterns within a full population or sample from a population.
Conclusion
In summary, understanding the population standard deviation (pop SD) is crucial for proper statistical analysis. The pop SD represents the spread or variability of values around the mean in an entire population. It allows researchers to determine how much sampling estimates like the sample SD may differ from the true pop SD.
Knowing the pop SD enables the calculation of confidence intervals, which indicate the reliability of sample statistics. It is also key for significance testing, as the pop SD is used to standardize test statistics. In essence, without understanding the inherent variability in a population as measured by pop SD, it would be impossible to draw valid conclusions from samples.
While samples provide insight into a population, only with knowledge of the population’s variability per the pop SD can researchers reliably generalize findings to the broader population. The pop SD sets the benchmark for determining if differences within samples reflect significant effects or random chance. Mastering this fundamental statistical measure is vital for anyone conducting robust quantitative analysis.