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.
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?
1 answer