Chapter 6 - Standard Deviation as a Ruler & Normal Model
| Monday, 1 January 0001 | |
| 2-minute read | |
| 282 words | |
Z-Score
\[ Z = \frac{x - \bar{x}}{S} \]
$x$ is the data value
$\bar{x}$ is the mean
$S$ is the standard deviation
The z-score is essentially a ratio of a given datapoint's deviation to the standard deviation.
This can only be applied to normally or approximately normally distributed data.
Example
At a selective college, two professors, Dr. V and Dr. W, have a competition to see who can give the most difficult tests in Calculus I. They compared their data from the first test of the semester.
| Professor | mean(x) | S |
|---|---|---|
| Dr. V | 47 | 18 |
| Dr. W | 51 | 12 |
Lyle is a student in Dr. V's class and he scared an 83. Herman is a student in Dr. W's class and he scored an 81. Who did better relative to the rest of his class?
Without knowing the shape or distribution of the data, we must use the z-score to determine who did better.
$Z_L = \frac{83 - 47}{18} = 2$
$Z_H = \frac{81 - 51}{12} = 2.5$
Herman did better relative to his class than Lyle did.
Empirical Rule and the Normal Model
Also known as the 68-95-99.7 rule.
The proportional area under the curve between $(a, b)$ determines the probability that any selected datapoint falls within the interval $(a, b)$
If the normal model does not mathematically make sense (normal model runs out of room, going to impossibly negative numbers), then the data is most likely not normally distributed.
normalcdf Example
Assuming the mean is 30, the standard deviation is 2, what percentage of the data is between 20 and 26?
$\textrm{normalcdf}(30, 2) = 0.02274984$
On a TI calculator:
normalcdf(lower, upper, mu, sigma)
On a Casio calculator:
NormCD(lower, upper, sigma, mu)