Normal distribution and z-scores

Definitions & Key takeaways

The normal distribution is a continuous probability distribution that is symmetric about the mean, with a bell-shaped curve. 68%, 95%, and 99% of the data lies within one, two, and three standard deviations from the mean, respectively. The normal distribution represents the occurrence of many natural phenomena.

On the other hand, a Z-score indicates the number of standard deviations between a certain value and the mean. A Z-score of 0 indicates that the data point is exactly at the mean. A Z-score of 1 indicates that the data point is one standard deviation above the mean, and a Z-score of -1 indicates that the data point is one standard deviation below the mean. Z-scores can be used to determine how unusual a data point is within a dataset that follows a normal distribution.

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Let’s say you ask 1000 men for their weight, and then you plot their answers on a histogram, which is a plot that shows the distribution of any measurement or data.
Let’s say that the average weight is 170 pounds or about 77 kilograms, and that it turns out that the majority of men weighed that amount, whereas fewer men weighed a little bit higher or a little bit lower than the average, and even fewer men weighed much higher or much lower than the average.
If we draw a curve over the top of our histogram, we get the normal distribution curve, which is also called the bell curve, because it’s shaped like a bell.
The bell curve is symmetrical, with half the data on the left of the average and half the data on the right side of the average.
The area under the bell curve is equal to 1, or 100%, with the highest percentage of data in the middle section and the lowest percentage of data in the outer tails of the curve.
Typically, for population data, the average point in a bell curve is labeled with the greek letter mu, and mu refers to the mean, median, and mode, because when data are normally distributed, the mean, median, and mode are all equal to each other.
The standard deviation is a measure of how spread out the data are from the average, and for population data it’s represented by the greek letter sigma.
For example, let’s say the standard deviation of weight for our sample of men is 29 pounds, or 13 kilograms. In a normal distribution, 68 percent of the data are within one standard deviation.
That means that 68 percent of men will weigh somewhere between 170 minus 29, or 141 pounds, and 170 plus 29, or 199 pounds.
Also, 95 percent of the data are found within two standard deviations - so, since 29 times 2 is 58, then 95 percent of men will weigh somewhere between 170 minus 58, or 112 pounds, and 170 plus 58, or 228 pounds.
Finally, 99.7 percent the data are found within three standard deviations, and since 29 times 3 is 87, 99.7% of men will weigh between 170 minus 87, or 83 pounds, and 170 plus 87, or 257 pounds.
This is called the empirical rule, or the 68-95-99.7 rule. Now, the shape of the bell curve depends on the size of the standard deviation.
A small standard deviation, like if it was only 5 pounds, tells you that most of the data are clustered around the average - and this makes the bell curve very tall and skinny.
On the other hand, a large standard deviation, like if it was 50 pounds, tells you that most of the data are way above and way below the average - and this makes the bell curve look very wide and flat.
Now, let’s say a man named Micah weighs 220 pounds, and he wants to know how close his weight is to the average weight. We can calculate how much more he weighs than the average by subtracting the average weight, 170 from his weight, 220, which equals 50.
But telling Micah that he weighs 50 pounds over the average doesn’t really have much meaning, because he probably doesn’t know if 50 pounds is a lot higher or only a little higher than the average.
Instead, we might tell Micah his z-score, or standard score, which is a measure of how many standard deviations his weight is from the average weight.
Z-scores range from negative 3 standard deviations, which would be on the very far end of the left tail, to positive 3 standard deviations, which would be on the very far end of the right tail.
In the normal distribution, the average value is the reference point, so the average value equals 0 standard deviations.
To figure out a z-score for an individual measurement - like Micah’s weight - we use the equation z equals the measurement minus the average measurement in the population, divided by the standard deviation for the population.
Usually, the individual measurement is represented by the letter x, so the equation can also be written z equals x minus mu, divided by sigma.
So, to figure out Micah’s z-score, we do 220 minus 170, divided by 29, which equals 1.72. This means that Micah weight is 1.72 standard deviations above the population average.
We can also figure out what percentage of men weighed more or less than Micah, by using a standard normal distribution table, or simply a z-score table.
The y-axis in the table represents the first two digits of the z-score - which in this case is 1.7 - and the x-axis in the table represents the third digit of the z-score - which is 0.02.
So, we find the row for 1.7 and the column for 0.02, and the number where they meet is the percentage of men who weigh less than 1.72 standard deviations.
Since the number is 0.9573, that means that 95.73 percent of men in the population will weigh less than 1.72 standard deviations.
In other words Micah weighs more than about 96% of the men. We can also calculate how many men will weigh more than Micah by subtracting the percentage from 100 - so 100 minus 95.73 is about 4%.
Z-score tables are also useful for figuring out the standard deviation if we have a certain percentage. For example, a person might find out that they’re in the top 10th percentile, which means that 10 percent of the population weighs more than that person and 90 percent of the population weighs less than that person.
To find out how many standard deviations above the average a person in the 10th percentile is, we work backwards using the z-score table.
So, we find the number in the table that is closest to 90%, or 0.90, which in this case is 0.8997. Then, we find the first two digits for that row, which is 1.2, and the third digit, which is 0.08, and add them together.
So 1.2 plus 0.08 is 1.28, which means that a person in the 90th percentile for weight is 1.28 standard deviations above the average.
And this number makes sense because of the empirical rule, since 1.28 is between 1 and 2 standard deviations, and 90 percent is between 68 and 95 percent.
Stated in terms of pounds, it would be 1.28 times the standard deviation of 29 pounds, so 37 pounds. So 170 pounds minus 37 pounds is 133 pounds, and that’s the 10 percent mark, and 170 pounds plus 37 pounds is 207 pounds, and that’s the 90% mark.
Now, it’s also possible that the population of 1000 men have a skewed distribution instead of a normal distribution, meaning one tail of the bell curve is longer than the other.
A right-skewed distribution means that the right tail is longer than the left tail, and a left-skewed distribution means that the left tail is longer than the right tail.
When the distribution is skewed, the mean, median, and mode are not equal to one another. And the mean isn’t in the middle of the curve, so the empirical rule doesn’t work.
For example, let’s say the population of men had a right-skewed distribution, so the lowest weight was 140 and the highest weight was 300, but the mean was still 170 and the standard deviation was 29.
In this case, it’s impossible to calculate 2 standard deviations below the mean, because 170 minus 58 is 112, and that’s out of the range of weights for this population.
So, data in a skewed distribution is usually described by its median, plus its first quartile - which is the middle number between the median and the smallest measurement - and third quartile - which is the middle number between the median and the largest measurement.
Alright, as a quick recap. The standard deviation shows how far away certain measurements are from the average, and in a normal distribution, 68 percent of the measurements are within 1 standard deviation of the average, 95 percent are within 2 standard deviations, and 99.7 percent are within 3 standard deviations.
The z-score equals x minus mu, divided by sigma, where x is the individual measurement, mu is the population average, and sigma is the standard deviation.
We can then look up what percentage of measurements are above or below that z-score using a z-score table.