如何使期望结果的概率最大化

In a topic as vast as statistics, we have learned so much from box and whisker plots to standard deviation to probability. Now, we will put together some of the concepts we have learned recently as we learn about z-scores.

##z-score

A z-score is a way of understanding where a data point lies in relation to the mean. In reference to the normal distribution, z-scores tell us how many standard deviations away a data point is from the mean.

There is a formula we can use to calculate the z-score:

$$z = \frac{x - \mu}{\theta}$$

Where...

x = the datapoint

$\mu$ = the population mean

$\theta$ = the standard deviation

Let's walk through an example of how to calculate a z-score.

Fiona just ran a 7 minute mile. She is trying to figure out how this time fits in with her other mile times. Her average mile time is 6.5 minutes with a standard deviation of 2. What is the z-score of her recent mile time?

We are given the mean and standard deviation, so we can solve using the z-score equation.

$$z = \frac{x - \mu}{\theta}$$ $$z = \frac{7 - 6.5}{2}$$ $$z = 0.25$$

Therefore, we can say that Fiona's 7 minute mile time is 0.25 standard deviations away from the mean.

Try an example on your own!

(Video) Statistics - Find the z score

by MySecretMathTutor

Now that we've learned about the normal distribution with its 3 Sigma Rule and skewness, it's time to learn about z-scores!

Check out this video by MySecretMathTutor to learn more!

Z-scores represent how many standard deviations above or below the mean a value is.

The formula to find the z-score is: $$z = \frac{x - \mu}{\theta}$$

where the $z$ stands for z-score, $\mu$ stands for mean, $\theta$ stands for standard deviation, and $x$ is the value we are finding the z-score for.

Let's walk through the first example in the video!

We are given that the mean is 24, the standard deviation is 5, and the value is 20. To find the z-score, we can plug these values into the equation above and solve for $z$. $z$ equals $-0.8$. Watch the video for step-by-step details!

Finding the Z-Score

The z-score is a raw score that tells you the quantifiable distance between a data point and the mean of a dataset. It is the number of standard deviations away from the mean a data point is. A positive z-score indicates that the point lies above the mean, while a negative z-score indicates that the point lies below the mean. The sample size must be above $30$ in order to use a z-score.

There is a simple formula to calculate the z-score, where you need to know the datapoint, population mean, and the standard deviation.

There is a table that you can use to find the probability, based on a standard normal distribution.