What is the z-score for 2 standard deviations?

Z-scores are expressed in terms of standard deviations from their means. Resultantly, these z-scores have a distribution with a mean of 0 and a standard deviation of 1. The formula for calculating the standard score is given below:

As the formula shows, the standard score is simply the score, minus the mean score, divided by the standard deviation. Therefore, let's return to our two questions.

1. How well did Sarah perform in her English Literature coursework compared to the other 50 students?

To answer this question, we can re-phrase it as: What percentage (or number) of students scored higher than Sarah and what percentage (or number) of students scored lower than Sarah? First, let's reiterate that Sarah scored 70 out of 100, the mean score was 60, and the standard deviation was 15 (see below).

In terms of z-scores, this gives us:

The z-score is 0.67 (to 2 decimal places), but now we need to work out the percentage (or number) of students that scored higher and lower than Sarah. To do this, we need to refer to the standard normal distribution table.

This table helps us to identify the probability that a score is greater or less than our z-score score. To use the table, which is easier than it might look at first sight, we start with our z-score, 0.67 (if our z-score had more than two decimal places, for example, ours was 0.6667, we would round it up or down accordingly; hence, 0.6667 would become 0.67). The y-axis in the table highlights the first two digits of our z-score and the x-axis the second decimal place. Therefore, we start with the y-axis, finding 0.6, and then move along the x-axis until we find 0.07, before finally reading off the appropriate number; in this case, 0.2514. This means that the probability of a score being greater than 0.67 is 0.2514. If we look at this as a percentage, we simply times the score by 100; hence 0.2514 x 100 = 25.14%. In other words, around 25% of the class got a better mark than Sarah (roughly 13 students since there is no such thing as part of a student!).

Going back to our question, "How well did Sarah perform in her English Literature coursework compared to the other 50 students?", clearly we can see that Sarah did better than a large proportion of students, with 74.86% of the class scoring lower than her (100% - 25.14% = 74.86%). We can also see how well she performed relative to the mean score by subtracting her score from the mean (0.5 - 0.2514 = 0.2486). Hence, 24.86% of the scores (0.2486 x 100 = 24.86%) were lower than Sarah's, but above the mean score. However, the key finding is that Sarah's score was not one of the best marks. It wasn't even in the top 10% of scores in the class, even though at first sight we may have expected it to be. This leads us onto the second question.

2. Which students came in the top 10% of the class?

A better way of phrasing this would be to ask: What mark would a student have to achieve to be in the top 10% of the class and qualify for the advanced English Literature class?

Although the finance industry can be complex, an understanding of the calculation and interpretation of fundamental mathematical building blocks is still the foundation for success, whether in accounting, economics, or investing.

Standard deviation and the Z-score are two such fundamentals. Z-scores can help traders gauge the volatility of securities. The score shows how far away from the mean—either above or below—a value is situated. Standard deviation is a statistical measure that shows how elements are dispersed around the average, or mean. Standard deviation helps to indicate how a particular investment will perform, so, it is a predictive calculation.

A firm grasp of how to calculate and utilize these two measurements enables a more thorough analysis of patterns and changes in any data set, from business expenditures to stock prices.

• Standard deviation defines the line along which a particular data point lies.
• Z-score indicates how much a given value differs from the standard deviation.
• The Z-score, or standard score, is the number of standard deviations a given data point lies above or below mean.
• Standard deviation is essentially a reflection of the amount of variability within a given data set.
• Bollinger Bands are a technical indicator used by traders and analysts to assess market volatility based on standard deviation.

The Z-score, or standard score, is the number of standard deviations a given data point lies above or below the mean. The mean is the average of all values in a group, added together, and then divided by the total number of items in the group.

To calculate the Z-score, subtract the mean from each of the individual data points and divide the result by the standard deviation. Results of zero show the point and the mean equal. A result of one indicates the point is one standard deviation above the mean and when data points are below the mean, the Z-score is negative.

In most large data sets, 99% of values have a Z-score between -3 and 3, meaning they lie within three standard deviations above or below the mean.

Z-scores offer analysts a way to compare data against a norm. A given company’s financial information is more meaningful when you know how it compares to that of other, comparable companies. 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.

A Bollinger Band® is a technical indicator used by traders and analysts to assess market volatility based on standard deviation. 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.