Describe, calculate, and interpret the Z-score. Explain how it relates to Standard deviation.
- Suppose you're given a data set. Perhaps you're even given many of the key statistical tools needed to start forming a statistical opinion, mean, median, range, standard deviation. By now, we should understand that these tools help us start asking important questions about our data set. We also learned that these tools help us in studying not just the entire data set, but also each individual value in the data set.
In particular, the standard deviation allows us to see whether or not individual data points might be considered outliers. Is your data point within one standard deviation of the mean, two standard deviations? How would I know if it is exactly 2.37 standard deviations from the median? Luckily, there's a very simple formula to help us find just how many standard deviations our data points lie from the mean.
This is what we call our Z-Score, and as I said, the formula to find your Z score is quite simple. As you can see, all you need is the data set's mean and standard deviation. Then, all you do is plug in one of the values in the data set. Let's plug in our largest value in the data set, 231. For this data set, we have a mean of 130.1 and a standard deviation of 47.85.
As we can see, we get a Z-Score of 2.11. That means the data point, 231, is 2.11 standard deviations from the mean in the positive direction. Let's do this same calculation for our lowest value. The only thing we change for this calculation is that we switch out 50 for 231. Notice what happens here. Now our Z-Score is negative 1.67.
This means that this data point is 1.67 standard deviations from our mean in the negative direction, which, of course, makes sense since 50 is well below our mean value, 130.1. Next time someone says that a certain outcome is 2.8 standard deviations from the mean, not only will you know what that means, you'll also know how it was calculated. Plus, knowing this simple formula will be helpful in determining whether an individual data point might be considered an outlier.
Professor Eddie Davila covers statistics basics, like calculating averages, medians, modes, and standard deviations. He shows how to use probability and distribution curves to inform decisions, and how to detect false positives and misleading data. Each concept is covered in simple language, with detailed examples that show how statistics are used in real-world scenarios from the worlds of business, sports, education, entertainment, and more. These techniques will help you understand your data, prove theories, and save time, money, and other valuable resources—all by understanding the numbers.
- Calculate mean and median for specific data sets.
- Explain how the mode is used to assess a data set.
- Identify situations in which standard deviation can be used to investigate individual data points.
- Use mean and standard deviation to find the Z-score for a data point.
- List the three different categories of probability.
- Analyze data to determine if two events are dependent or independent.
- Predict possible outcomes for a situation using basic permutation calculations.
- Give examples of binomial random variables.