A business has just completed a quarterly performance audit and the manager has been informed that they received a score of 55. How should this manager feel?
What information might the manager want to know? Perhaps he’d like to know the average (mean) score of all audits for this quarter. If he were to discover it was 50, he might feel better about his performance.
A manager might be interested in knowing the distribution of the scores above and below the mean. Were the majority of the scores close to the mean, or far above or below the mean? We can infer on the dispersion of scores with the basic range (low score minus high score). If the range was 30 points, with the high limit at 75 and the low limit at 45, the manager might be disappointed with their score of 55 even though it was above the average.
An inherent vulnerability with using range is that all it takes is one extreme score to distort it’s meaning. In the previous example consider that only one business scored a 75 with the next highest being 56 with everyone else falling between 46 and 56 (range of 10).
Standard deviation is a better choice when examining the dispersion of scores. Let’s say instead that the manager was told that 68% of businesses score between 48 and 52. In other words, 68% of business lay 2 points above or below the mean. This would mean the audit scores had a standard deviation of 2. If we were to assume normality (this is dangerous and I’ll explain in my next post) that is to say the scores fell into a predictable normal distribution, we could infer that 95% of the businesses would have scores within two standard deviations above or below the mean. In our case that would 4 points above (54) or 4 points below (46) the mean (50). The manager would feel good about his score of 55. Knowing the mean is not enough, but knowing the standard deviation along side it, paints a different story. It allows one to interpret raw scores and compare businesses’ performance to each other or the same business over different audits.
standardize |ˈstandərˌdīz| -determine the properties of by comparison with a standard.
Now that I’ve mentioned standard deviations, standardized scores will be intuitive. Consider an audit score of 30 for quarter 1 and a score of 125 for quarter 2. The only interpretation we can make without understanding their relationship to the mean is that a business improved from quarter 1 to 2. Unlike my example in the previous paragraph, being told that the Q1 score is 2 points above the mean and the Q2 score is 10 points above the mean doesn’t offer any additional meaning. Perhaps each audit is different as the audit procedure was updated to reflect new products or additional equipment to inspect. With this in mind being 2 points above the mean may be better than being 10 points above the mean on another test. A standard score will enable one to compare scores that come from different sources.
The most common standard score is the z-score, it describes how many standard deviations a business’ score is from the mean. For example a z-score of -1.4 indicates that a business is 1.4 standard deviations below the mean.
Calculating a z-score is easy: subtract the mean from the raw score and divide by the standard deviation. For example: raw score = 15, mean = 10, standard deviation = 4. Therefore 15 minus 10 equals 5. 5 divided by 4 equals 1.25. Thus the z-score is 1.25.
Below is an illustration I made to exhibit how extreme values and outliers can affect the interpretation of “average”.