In a certain school district in a large metropolitan area, the SAT scores over that past five years are normally distributed with a mean of 1476. Furthermore, P20
is 1209.
Using this information, what is the standard deviation of this population of SAT scores?
In statistics, when dealing with normally distributed data, the percentile (also referred to as a quantile) of a distribution is the value below which a given percentage of observations in a group of observations falls. For example, P20 (the 20th percentile) is the value below which 20% of the observations may be found.
The question provides us with the following information:
- Mean (μ) = 1476
- 20th percentile (P20) = 1209
We are asked to find the standard deviation (σ) of this population.
The standard deviation can be estimated using the Z-score formula for a normal distribution, which is:
Z = (X - μ) / σ
where Z is the Z-score corresponding to a given percentile, X is the score corresponding to the percentile, μ is the mean, and σ is the standard deviation.
First, we must find the Z-score that corresponds to the 20th percentile for a normal distribution. This can be found using a Z-table, or using technology (like statistical software or a calculator with statistical functions).
Looking up the Z-score for the 20th percentile in a standard normal distribution table gives us approximately -0.8416. This value of Z indicates how many standard deviations an observation (in this case, the SAT score that corresponds to the 20th percentile) is from the mean.
Now we place the values into the Z-score formula:
-0.8416 = (1209 - 1476) / σ
We are solving for σ, so let's do the arithmetic inside the parentheses first:
-0.8416 = -267 / σ
Now we solve for σ by multiplying both sides by σ and then dividing both sides by -0.8416:
σ = -267 / -0.8416
σ ≈ 317.18
Therefore, the standard deviation of the population of SAT scores is approximately 317.18.