Standard deviation

Question

The population parameters below describe the full-time equivalent number of students (FTES) each year at Lake Tahoe Community College from 1976-1977 through 2004-2005.

•	𝜇 = 1000 FTES
•	median = 1,014 FTES
•	𝜎 = 474 FTES
•	first quartile = 528.5 FTES
•	third quartile = 1,447.5 FTES
•	n = 29 years

What is the population standard deviation?

Answer 1

μ (which we say "myoo") and σ (which we say "sigma") are letters of the Greek alphabet. Statisticians use Greek letters when describing a population parameter. A parameter is either a measure of center or a measure of variation in a population.

If you were to place all 29 values of the FTES on a number line, then the "mean" μ is right at the center of all of the values. If our data is symmetric, then it should work out that half the FTES numbers should be to the right of the mean, and half the numbers should be to the left of it. None of this is precise. It only has to be close to half. This number is used as a basis of comparison for any other number in our data. How far, or how close, to the mean is a given year means that the FTES numbers are either stable, or not stable. Predictable, or not predictable. Does this idea make sense?

Enter onto the stage... "standard deviation".

The Greek letter σ is used to describe how spread out the population is. By "population" we mean the 29 years of enrollment at LTCC. The definition of sigma (σ) is "population standard deviation". This tells us a lot about enrollment at LTCC. Does it seem to fluctuate a lot? Or is it fairly predictable and steady?

The mean (μ) is given to us as 1,000. σ = 474. What this means is that about 68% of 29 values lie between 1000 - 474 and 1000 + 474. Doing the math, we see that 68% of the time, FTES will be between 526 and 1,474. (Seems like a fairly wide range to me, does it to you as well?)

If we want to see where 95% of the numbers are, then we need to widen the range to 2 standard deviations, which means 2 * σ = 948 away from the mean.

The more we want to include, the wider the interval. So 95% of the time, we will see FTES between 1000 - 948 and 1000 + 948. In other words, FTES is largely between 52 students and 1,948 students. So if you had all 29 numbers on a number line spread out on a wall and you aimed a dart at the wall randomly, 95% of the time you'd hit near a number between 52 and 1,948. That isn't saying much, is it?