And now I want to add the, what I get on the 5th exam, x. And I'm going to divide that by all five exams. So in other words, this number is the average of my first five exams. We just figured out the average of the first four exams. But now, we sum up the first four exams here. We add what I got on the fifth exam, and then we divide it by 5, because now we're averaging five exams.
And I said that I need to get in an 88 in the class. And now we solve for x. Let me make some space here. So, 5 times 88 is, let's see. Well, it turns out if you subtract from both sides, you get x is equal to So unless you have a exam that has some bonus problems on it, it's probably impossible for you to get ah an 88 average in the class after just the next exam.
You'd have to get on that next exam. And let's just look at what we just did. We said, after 4 exams we had an What do I have to get on that next exam to average an 88 in the class after 5 exams?
And that's what we solved for when we got x. Now, let's ask another question. I said after four exams, after four exams, I had an 84 average. If I said that there are 6 exams in the class, and the highest score I could get on an exam is , what is the highest average I can finish in the class if I were to really study hard and get on the next 2 exams?
Well, once again, what we'll want to do is assume we get on the next 2 exams and then take the average. So we'll have to solve all 6 exams.
So we're going to have the average of 6, so in the denominator we're going to have 6. The first four exams, the sum, as we already learned, is 4 exams times the 84 average. And this dot is just times. Plus, and there's going to be 2 more exams, right? Because there's 6 exams in the class. And I'm going to get in each. So that's And what's this average? Well, 4 times 84, we already said, is Plus over 6.
So that's over 6. A resistant statistic is a numerical summary wherein extreme numbers do not have a substantial impact on its value. For instance, 10 people are having dinner at a restaurant. The table below shows their income from lowest to highest. The mean income and the range of the group is now too high.
Likewise, we can say Bill Gates is an outlier with an annual income that hits billions. This example shows that the mean and range are not resistant to extreme values.
While the median, as a numerical summary, generally exhibits resistance. What does this tell us? The presence of extreme values or outliers indicate that a distribution is skewed. Extreme values typically pull the mean toward the direction of the tail. If the scores tend toward the higher side of the scale, with few low scores , the distribution is negatively skewed. In finance, investors take note of skewness when they analyze return distribution. This is important because it allows them to see the extreme ranges of the data instead of just focusing on the average values.
A distribution shows skewness degree of asymmetry or kurtosis when the returns fall outside the normal distribution. Kurtosis measures the outliers in either tail of a skewed graph. It calculates the degree to which a graph is peaked compared to a normal distribution. How does it help investors? Observing skewness or kurtosis helps analysts predict risks that that result when a model following normal distribution is compared to a data set with a tendency for higher standard deviation.
The risk is determined by calculating how far the numbers are from the normal distribution. In statistics, outliers or anomalies are unusual observations that do not belong to a certain population.
Researchers commonly find outliers based on large, well-structured data. How different should a value be to be considered an outlier? To determine this, you can use the interquartile range IQR. The interquartile range IQR is also similar to range but is considered a less sensitive to extreme values resistant statistic. To find it, you must take the first quartile and subtract the third quartile. This shows how data is spread around the median.
Practically all sets of data can be described by the 5 number summary. IQR x 1. So far, no value is less than -3 or greater than 21 in the set. Though the maximum value 18 is 5 points more than 13, it is not considered an outlier for this data set. In sports analytics , researchers gather statistics to measure the potential and ability of professional athletes. According to Competitive Edge Athletic Performance Center , sports performance metrics are relevant to overall athletic development.
To achieve success in any sports field, individuals must reach certain levels of athleticism to compete at advanced levels.
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