Why z scores are useful

We may want to tell other people how far away a particular score is from average. We might also want to compare scores from different bunches of data. We will want to know which score is better. Z-scores can help with all of this. Z-Scores tell us whether a particular score is equal to the mean, below the mean or above the mean of a bunch of scores.

They can also tell us how far a particular score is away from the mean. In the next academic year, he must choose which of his students have performed well enough to be entered into an advanced English Literature class. He decides to use the coursework scores as an indicator of the performance of his students. Therefore, we are left with two questions to answer. First, how well did Sarah perform in her English Literature coursework compared to the other 50 students? Whilst it is possible to calculate the answer to both of these questions using the existing mean score and standard deviation, this is very complex.

Therefore, statisticians have come up with probability distributions , which are ways of calculating the probability of a score occurring for a number of common distributions, such as the normal distribution.

In our case, we make the assumption that the students' scores are normally distributed. As such, we can use something called the standard normal distribution and its related z-scores to answer these questions much more easily. When a frequency distribution is normally distributed, we can find out the probability of a score occurring by standardising the scores, known as standard scores or z scores. Z-score results of zero indicate that the data point being analyzed is exactly average, situated among the norm.

A score of 1 indicates that the data are one standard deviation from the mean, while a Z-score of -1 places the data one standard deviation below the mean. The higher the Z-score, the further from the norm the data can be considered to be. In investing, when the Z-score is higher it indicates that the expected returns will be volatile, or are likely to be different from what is expected.

Simply put, they are a visual representation of the Z-score. For any given price, the number of standard deviations from the mean is reflected by the number of Bollinger Bands between the price and the exponential moving average EMA. Standard deviation is essentially a reflection of the amount of variability within a given data set.

It shows the extent to which the individual data points in a data set vary from the mean. In investing, a large standard deviation means that more of your data points deviate from the norm, so the investment will either outperform or underperform similar securities. A small standard deviation means that more of your data points are clustered near the norm and returns will be closer to the expected results. Investors expect a benchmark index fund to have a low standard deviation.

However, with growth funds, the deviation should be higher as the management will make aggressive moves to capture returns. As with other investments, higher returns equate to higher investment risks.

The standard deviation can be visualized as a bell curve , with a flatter, more spread-out bell curve representing a large standard deviation and a steep, tall bell curve representing a small standard deviation. To calculate the standard deviation, first, calculate the difference between each data point and the mean. The differences are then squared, summed, and averaged to produce the variance. The standard deviation, then, is the square root of the variance, which brings it back to the original unit of measure.

In investing, standard deviation and the Z-score can be useful tools in determining market volatility. As the standard deviation increases, it indicates that price action varies widely within the established time frame.

Given this information, the Z-score of a particular price indicates how typical or atypical this movement is based on previous performance.

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