How do you calculate standard deviation in python?
Two closely related statistical measures will allow us to get an idea of the spread or dispersion of our data. The first measure is the variance, which measures how far from their mean the individual observations in our data are. The second is the standard deviation, which is the square root of the variance and measures the amount of variation or dispersion of a dataset.
In this tutorial, we'll learn how to calculate the
variance and the standard deviation in Python. We'll first code a Python function for each measure and later, we'll learn how to use the Python
With this knowledge, we'll be able to take a first look at our datasets and get a quick idea of the general dispersion of our data.
Calculating the Variance
In statistics, the variance is a measure of how far individual (numeric) values in a dataset are from the mean or average value. The variance is often used to quantify spread or dispersion. Spread is a characteristic of a sample or population that describes how much variability there is in it.
A high variance tells us that the values in our dataset are far from their mean. So, our data will have high levels of variability. On the other hand, a low variance tells us that the values are quite close to the mean. In this case, the data will have low levels of variability.
To calculate the variance in a dataset, we first need to find the difference between each individual value and the mean. The variance is the average of the squares of those differences. We can express the variance with the following math expression:
In this equation, xi stands for individual values or observations in a dataset. μ stands for the mean or average of those values. n is the number of values in the dataset.
The term xi - μ is called the deviation from the mean. So, the variance is the mean of square deviations. That's why we denoted it as σ2.
Say we have a dataset [3, 5, 2, 7, 1, 3]. To find its variance, we need to calculate the mean which is:
Then, we need to calculate the sum of the square deviation from the mean of all the observations. Here's how:
To find the variance, we just need to divide this result by the number of observations like this:
That's all. The variance of our data is 3.916666667. The variance is difficult to understand and interpret, particularly how strange its units are.
For example, if the observations in our dataset are measured in pounds, then the variance will be measured in square pounds. So, we can say that the observations are, on average, 3.916666667 square pounds far from the mean 3.5. Fortunately, the standard deviation comes to fix this problem but that's a topic of a later section.
If we apply the concept of variance to a dataset, then we can distinguish between the sample variance and the population variance. The population variance is the variance that we saw before and we can calculate it using the data from the full population and the expression for σ2.
The sample variance is denoted as S2 and we can calculate it using a sample from a given population and the following expression:
This expression is quite similar to the expression for calculating σ2 but in this case, xi represents individual observations in the sample and X is the mean of the sample.
S2 is commonly used to estimate the variance of a population (σ2) using a sample of data. However, S2 systematically underestimates the population variance. For that reason, it's referred to as a biased estimator of the population variance.
When we have a large sample, S2 can be an adequate estimator of σ2. For small samples, it tends to be too low. Fortunately, there is another simple statistic that we can use to better estimate σ2. Here's its equation:
This looks quite similar to the previous expression. It looks like the squared deviation from the mean but in this case, we divide by n - 1 instead of by n. This is called Bessel's correction. Bessel's correction illustrates that S2n-1 is the best unbiased estimator for the population variance. So, in practice, we'll use this equation to estimate the variance of a population using a sample of data. Note that S2n-1 is also known as the variance with n - 1 degrees of freedom.
Now that we've learned how to calculate the variance using its math expression, it's time to get into action and calculate the variance using Python.
Coding a variance() Function in Python
To calculate the variance, we're going to code a Python function called
Here's a possible implementation for
We first calculate the number of observations (
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The next step is to calculate the square deviations from the mean. To do that, we use a
Finally, we calculate the variance by summing the deviations and dividing them by the number of observations
In this case,
We can refactor our function to make it more concise and efficient. Here's an example:
In this case, we remove some intermediate steps and temporary variables like
Note that this implementation takes a second argument called
Using Python's pvariance() and variance()
includes a standard module called
Here's how Python's
We just need to import the
On the other hand, we can use Python's
This is the sample variance S2. So, the result of using Python's
Calculating the Standard Deviation
The standard deviation measures the amount of variation or dispersion of a set of numeric values. Standard deviation is the square root of variance σ2 and is denoted as σ. So, if we want to calculate the standard deviation, then all we just have to do is to take the square root of the variance as follows:
Again, we need to distinguish between the population standard deviation, which is the square root of the population variance (σ2) and the sample standard deviation, which is the square root of the sample variance (S2). We'll denote the sample standard deviation as S:
Low values of standard deviation tell us that individual values are closer to the mean. High values, on the other hand, tell us that individual observations are far away from the mean of the data.
Values that are within one standard deviation of the mean can be thought of as fairly typical, whereas values that are three or more standard deviations away from the mean can be considered much more atypical. They're also known as outliers.
Unlike variance, the standard deviation will be expressed in the same units of the original observations. Therefore, the standard deviation is a more meaningful and easier to understand statistic. Retaking our example, if the observations are expressed in pounds, then the standard deviation will be expressed in pounds as well.
If we're trying to estimate the standard deviation of the population using a sample of data,
then we'll be better served using n - 1 degrees of freedom. Here's a math expression that we typically use to estimate the population variance:
Coding a stdev() Function in Python
To calculate the standard deviation of a dataset, we're going to rely on our
If we want
With this new implementation, we can use
Using Python's pstdev() and stdev()
Here's how these functions work:
We first need to import the
If we don't have the data for the entire population, which is a common
scenario, then we can use a sample of data and use
The variance and the standard deviation are commonly used to measure the variability or dispersion of a dataset. These statistic measures complement the use of the mean, the median, and the mode when we're describing our data.
In this tutorial, we've learned how to calculate the variance and the standard deviation of a dataset using Python. We first learned, step-by-step, how to create our own functions to compute them, and later we learned how to use the Python
How do you find the standard deviation of a column in Python?
How to Calculate Standard Deviation in Pandas (With Examples).
Method 1: Calculate Standard Deviation of One Column df['column_name']. std().
Method 2: Calculate Standard Deviation of Multiple Columns df[['column_name1', 'column_name2']]. std().
Method 3: Calculate Standard Deviation of All Numeric Columns df. std().
What is the formula to calculate standard deviation?
The standard deviation formula may look confusing, but it will make sense after we break it down. ... .
Step 1: Find the mean..
Step 2: For each data point, find the square of its distance to the mean..
Step 3: Sum the values from Step 2..
Step 4: Divide by the number of data points..
Step 5: Take the square root..
What is STD function in Python?
std(arr, axis = None) : Compute the standard deviation of the given data (array elements) along the specified axis(if any).. Standard Deviation (SD) is measured as the spread of data distribution in the given data set. For example : x = 1 1 1 1 1 Standard Deviation = 0 .
How is std error calculated in Python?
How to Calculate the Standard Error of the Mean in Python.
Standard error of the mean = s / √n..
The larger the standard error of the mean, the more spread out values are around the mean in a dataset..
As the sample size increases, the standard error of the mean tends to decrease..