Statistics Notation: Mean, Variance & Normal Distribution

In summary: Thanks for catching that!Yes. The notation did use a \sigma^2 so the 9 is the variance, not the standard deviation. Thanks for catching that!
  • #1
ampakine
60
0
In my lecture notes on confidence intervals the lecturer wrote this:
Recall that measurements tend to follow a normal distribution. To describe the normal distribution and answer useful questions (as in the previous chapter), we need to know two numbers; the expectation or mean μ and the standard deviation (square root of the variance) σ. Then the quantity we measure X follows the normal distribution:
X ~ N(μ, σ2)

I don't understand the notation of that bolded text. I know X is a random variable, μ is the population mean and σ is the population standard deviation but what does the ~ mean? Also the N(μ, σ2) I assume means normal distribution but is that some kind of standard notation for distributions? For example if I said N(23,9) would that mean the normal distribution with a mean of 23 and standard deviation of 9?
 
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  • #2
ampakine said:
In my lecture notes on confidence intervals the lecturer wrote this:


I don't understand the notation of that bolded text. I know X is a random variable, μ is the population mean and σ is the population standard deviation but what does the ~ mean?

In probability theory, the notation is used to state the distribution for a random variable. You may think of the "~" as saying "distributed as" or "with distribution" or "has distribution".

Also the N(μ, σ2) I assume means normal distribution but is that some kind of standard notation for distributions?

It's a standard notation for normal distributions.

For example if I said N(23,9) would that mean the normal distribution with a mean of 23 and standard deviation of 9?

Yes
 
  • #3
That clears it up. Thanks!
 
  • #4
ampakine said:
For example if I said N(23,9) would that mean the normal distribution with a mean of 23 and standard deviation of 9?

Just to say it would be a normal with a mean of 23 and a variance of 9.
 
  • #5
QuendeltonPG said:
Just to say it would be a normal with a mean of 23 and a variance of 9.

Yes. My mistake. The notation did use a [tex] \sigma^2 [/tex] so the 9 is the variance, not the standard deviation.
 

What is the mean in statistics notation?

The mean, denoted by μ, is a measure of central tendency in statistics that represents the average value of a dataset. It is calculated by summing all the values in a dataset and dividing by the total number of values.

What is variance in statistics notation?

Variance, denoted by σ², is a measure of spread or variability in a dataset. It measures how far a set of numbers is spread out from their average value (mean). A smaller variance indicates that the data points are closer to the mean, while a larger variance indicates a wider spread of data points.

What is the normal distribution in statistics notation?

The normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetrical and bell-shaped. It is often used to model real-world phenomena and is characterized by its mean (μ) and standard deviation (σ). The notation for a normal distribution is N(μ, σ²), where μ represents the mean and σ² represents the variance.

How do you calculate the mean and variance in statistics notation?

To calculate the mean, you add up all the values in a dataset and divide by the total number of values. The formula for mean is: μ = Σx/n, where Σx represents the sum of all values and n represents the total number of values.

To calculate the variance, you first find the mean, then subtract each value from the mean, square the differences, and finally find the average of the squared differences. The formula for variance is: σ² = Σ(x-μ)²/n, where Σ represents the sum, x represents each value, μ represents the mean, and n represents the total number of values.

Why is the normal distribution important in statistics?

The normal distribution is important in statistics because it is commonly found in many real-world phenomena and can be used to model and analyze data. It also has many useful properties, such as the 68-95-99.7 rule, which states that approximately 68% of values fall within one standard deviation of the mean, 95% fall within two standard deviations, and 99.7% fall within three standard deviations. This makes it a useful tool for making predictions and drawing conclusions about data.

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