Lesson Video: Standard Deviation of Discrete Random Variables Mathematics • Third Year of Secondary School

Video Transcript

In this video, we will learn how to calculate the standard deviation of discrete random variables.

Standard deviation of a random variable is a measure of spread of the probability distribution. Given a random variable 𝑋, the standard deviation is denoted 𝜎 or 𝜎 sub 𝑋. Its square, which is called the variance, or var of 𝑋, is defined as follows: 𝜎 squared, or the variance of 𝑋, is equal to the 𝐸 of 𝑋 minus 𝐸 of 𝑋 all squared, where 𝐸 of 𝑋 denotes the expected value of the random variable 𝑋. The standard deviation 𝜎 is therefore obtained by taking the positive square root of the variance.

Looking at this formula a little closer, we see that the variance of 𝑋 is the average value of the square distance of data points from the expected value. In short, the standard deviation represents how far, on average, outcomes of the random variable are from the expected value. This can be demonstrated graphically. In the picture shown, the probability distribution of the random variable 𝑋 is given where 𝐸 of 𝑋 denotes the expected value or mean and 𝜎 denotes the standard deviation. This formula is cumbersome to use in practice, so we introduce a variant of this formula, which we will see shortly. Since this alternative formula is simpler to use, we will use it instead as the definition.

Given a random variable 𝑋, the variance of 𝑋 is given by var of 𝑋 is equal to 𝐸 of 𝑋 squared minus 𝐸 of 𝑋 all squared where 𝐸 of 𝑋 is the expected value or mean. The standard deviation 𝜎 or 𝜎 sub 𝑋 can therefore be calculated by square rooting this variance of 𝑋. The process of computing the standard deviation of a discrete random variable can therefore be summarized in four steps.

Step one is to compute 𝐸 of 𝑋. For any discrete random variable 𝑋, taking values 𝑥 sub one, 𝑥 sub two, and so on up to 𝑥 sub 𝑛, the expected value is equal to 𝑥 sub one multiplied by the probability that 𝑋 equals 𝑥 sub one plus 𝑥 sub two multiplied by the probability that 𝑋 equals 𝑥 sub two and so on all the way up to 𝑥 sub 𝑛 multiplied by the probability that 𝑋 equals 𝑥 sub 𝑛. Our second step is to compute 𝐸 of 𝑋 squared. This can be calculated in a similar way as 𝐸 of 𝑋, except we use 𝑥 sub one squared, 𝑥 sub two squared, and so on. We square the individual 𝑥-values before multiplying them by the corresponding probabilities.

Our third step is to compute the variance of 𝑋. We do this using the formula above. The variance of 𝑋 is equal to 𝐸 of 𝑋 squared minus 𝐸 of 𝑋 all squared. Finally, we’re in a position to compute the standard deviation 𝜎 by square rooting the variance of 𝑋. We will now look at some examples where we need to follow these four steps in different contexts.

The function in the table is a probability function of a discrete random variable 𝑋 find the standard deviation of 𝑋. Give your answer to two decimal places.

In order to answer this question, we recall the four-step process used to obtain the standard deviation 𝜎. Firstly, we compute the mean or expected value 𝐸 of 𝑋. Secondly, we compute 𝐸 of 𝑋 squared. Thirdly, we compute the variance or var of 𝑋. This is equal to the 𝐸 of 𝑋 squared minus the 𝐸 of 𝑋 all squared. Our fourth and final step is to compute the standard deviation 𝜎, which we recall is equal to the square root of the var of 𝑋.

Let’s begin by recalling how we calculate the 𝐸 of 𝑋. We do this by multiplying each value of 𝑋 by its corresponding probability and then finding the sum of these values. We multiply negative five by one-third. We then add the product of negative four and one-eighth, negative three and one-quarter, and finally negative one and seven twenty-fourths. Whilst we could work out each of the four products individually, typing the whole calculation into our calculator gives us negative 77 over 24.

Our second step is to calculate 𝐸 of 𝑋 squared. In order to do this, it is worth adding an extra row to our table to calculate the 𝑋 squared values. Negative five squared is 25, as multiplying a negative number by a negative number gives a positive answer. In the same way, squaring negative four, negative three, and negative one gives us values of 16, nine, and one. We can now repeat the process we used to calculate 𝐸 of 𝑋. This time, we multiply the 𝑋 squared values by their corresponding probabilities. This gives us 25 multiplied by one-third plus 16 multiplied by one-eighth plus nine multiplied by one-quarter plus one multiplied by seven twenty-fourths. Once again, we can type this directly into our calculator, giving us 103 over eight.

The third step of our process is to calculate the variance or var of 𝑋. This is equal to 𝐸 of 𝑋 squared minus 𝐸 of 𝑋 all squared. Substituting in the values we have calculated, we have 103 over eight minus negative 77 over 24 squared. Typing this into our calculator gives us 1487 over 576. As the standard deviation is the square root of the variance, we can calculate this by square rooting 1487 over 576. Noting that we need to give our answer to two decimal places, this is approximately equal to 1.61. The standard deviation of our function to two decimal places is 1.61.

In our next question, one of the values of 𝑋 in our table will be unknown.

The function in the given table is a probability function of a discrete random variable 𝑋. Given that the expected value of 𝑋 is 6.5, find the standard deviation of 𝑋. Give your answer to two decimal places.

In this question, we are given the expected value 𝐸 of 𝑋, which is equal to 6.5. We can use this to help us identify the unknown parameter 𝐴. We recall that we can calculate the expected value by multiplying each of our 𝑋-values by the corresponding probabilities. We then find the sum of all these values. This means that 𝐸 of 𝑋 is equal to three multiplied by 0.2 plus 𝐴 multiplied by 0.1 plus six multiplied by 0.1 plus eight multiplied by 0.6. Simplifying this right-hand side, we have 0.6 plus 0.1𝐴 plus 0.6 plus 4.8. And we know this is equal to 6.5. Subtracting 0.6, 0.6, and 4.8 from both sides of our equation gives us 0.5 is equal to 0.1𝐴. We can then divide both sides of this equation by 0.1 giving us 𝐴 is equal to five. The missing value in our table is five such that the probability that 𝑋 equals five is 0.1.

Next, we recall that to compute the standard deviation, we need to follow four steps. Firstly, we compute 𝐸 of 𝑋. Secondly, we compute 𝐸 of 𝑋 squared. Our third step is to compute the variance or var of 𝑋 which is equal to 𝐸 of 𝑋 squared minus the 𝐸 of 𝑋 all squared. Finally, we can compute the standard deviation 𝜎 by square rooting the variance of 𝑋.

Clearing some space, we already know that the expected value or mean 𝐸 of 𝑋 is equal to 6.5. We calculate 𝐸 of 𝑋 squared in a similar way to 𝐸 of 𝑋. This is equal to three squared multiplied by 0.2 plus five squared multiplied by 0.1 plus six squared multiplied by 0.1 plus eight squared multiplied by 0.6. Typing this into our calculator gives us 46.3. We now have values of both 𝐸 of 𝑋 and 𝐸 of 𝑋 squared. The var of 𝑋 is equal to the 𝐸 of 𝑋 squared minus the 𝐸 of 𝑋 all squared. So, in this case, we have 46.3 minus 6.5 squared. This is equal to 4.05. Finally, we can calculate the standard deviation by square rooting this variance. To two decimal places, this is equal to 2.01. The standard deviation of the function in the given table to two decimal places is 2.01.

Before looking at one final example, we will consider the coefficient of variation. The coefficient of variation, written 𝐶 sub 𝑉, gives the standard deviation as a percentage of the expected value. If we let 𝑋 be a discrete random variable with mean 𝐸 of 𝑋 and standard deviation 𝜎 sub 𝑋, if we assume further that 𝜇 is not equal to zero, then the coefficient of variation 𝐶 sub 𝑉 is given by 𝐶 sub 𝑉 of 𝑋 is equal to 𝜎 sub 𝑋 divided by 𝐸 of 𝑋 multiplied by 100. We assume that 𝜇 is not equal to zero as 𝐶 sub 𝑉 is not defined when the mean equals zero. As the standard deviation is always positive, the coefficient of variation will be negative when 𝐸 of 𝑋 is negative and positive when 𝐸 of 𝑋 is positive.

It is important to note that while the standard deviation is an absolute measure of spread, the coefficient of variation is a relative measure of spread. This is useful as when we deal with variables with larger expected values, they’re more likely to be more spread out. It therefore makes sense to use a relative measure when comparing spreads. The coefficient of variation is also useful when comparing data sets with different means and standard deviations. The coefficient of variation therefore represents how far on average data points are from the mean relative to the size of the mean. We will now look at an example where we need to calculate this coefficient of variation.

Work out the coefficient of variation of the random variable 𝑋 whose probability distribution is shown. Give your answer to the nearest percent.

We know that our figure is a probability distribution graph. And we recall that the coefficient of variation, written 𝐶 sub 𝑉, is equal to the standard deviation 𝜎 divided by the expected value or mean 𝐸 of 𝑋 multiplied by 100 percent. This coefficient of variation represents how far on average data points are from the mean relative to the size of the mean. We will begin by calculating the mean or expected value 𝐸 of 𝑋. We do this by multiplying each of our 𝑋-values by the corresponding 𝑓 of 𝑥 value or probability. We then find the sum of all these products.

From the graph, we begin by multiplying one by one-tenth. Next, we multiply three by two-tenths. We also need to multiply five by three-tenths and seven by four-tenths. Calculating each of these products gives us 0.1, 0.6, 1.5, and 2.8. 𝐸 of 𝑋 is therefore equal to five. As we also need to calculate the standard deviation, our next step is to calculate 𝐸 of 𝑋 squared. This is equal to one squared multiplied by one-tenth plus three squared multiplied by two-tenths plus five squared multiplied by three-tenths plus seven squared multiplied by four-tenths. This is equal to 0.1 plus 1.8 plus 7.5 plus 19.6. 𝐸 of 𝑋 squared is therefore equal to 29.

Next, we recall that the variance or var of 𝑋 is equal to 𝐸 of 𝑋 squared minus 𝐸 of 𝑋 all squared. In this question, we have 29 minus five squared. This is equal to four. Clearing some space, we have the following three values. We know that the standard deviation 𝜎 is equal to the positive square root of the variance of 𝑋. This means that in this question, the standard deviation is the positive square root of four, which equals two. We can now substitute our values into the formula for the coefficient of variation. We need to multiply two-fifths or 0.4 by 100. This is equal to 40 percent. The coefficient of variation of the random variable 𝑋 shown in the graph is 40 percent.

We will now finish this video by summarizing the key points. Given the probability distribution of a random variable 𝑋, we can compute the standard deviation 𝜎 using the following steps. (i) Compute the mean or expected value 𝐸 of 𝑋, (ii) compute 𝐸 of 𝑋 squared, (iii) compute the variance, the var, of 𝑋, which is equal to 𝐸 of 𝑋 squared minus 𝐸 of 𝑋 all squared, and (iv) compute 𝜎 the standard deviation by finding the positive square root of the var of 𝑋.

We also saw that the coefficient of variation, 𝐶 sub 𝑉, represents the standard deviation 𝜎 as a percentage of 𝐸 of 𝑋, the expected value, such that 𝐶 sub 𝑉 is equal to 𝜎 divided by 𝐸 of 𝑋 multiplied by 100 percent. We note that standard deviation is an absolute measure of spread, and the coefficient of variation is a relative measure of spread.