Calculating standard scores 6 A set of scores has a mean of 100 and a standard deviation of 20 . Calculate standardised scores for each of the following test scores: \( \begin{array}{lllll}\text { a } 120 & \text { b } 140 & \text { c } 80 & \text { d } 100 & \text { e } 40\end{array} \)

To calculate the standardized scores (also known as **z-scores**) for each test score, you can use the following formula: \[ z = \frac{{x - \mu}}{{\sigma}} \] where: - \( x \) = individual test score - \( \mu \) = mean of the scores - \( \sigma \) = standard deviation Given: - Mean (\( \mu \)) = 100 - Standard Deviation (\( \sigma \)) = 20 Let's calculate the z-scores for each test score: ### a) Test Score = 120 \[ z = \frac{{120 - 100}}{{20}} = \frac{{20}}{{20}} = **1.0** \] ### b) Test Score = 140 \[ z = \frac{{140 - 100}}{{20}} = \frac{{40}}{{20}} = **2.0** \] ### c) Test Score = 80 \[ z = \frac{{80 - 100}}{{20}} = \frac{{-20}}{{20}} = **-1.0** \] ### d) Test Score = 100 \[ z = \frac{{100 - 100}}{{20}} = \frac{{0}}{{20}} = **0.0** \] ### e) Test Score = 40 \[ z = \frac{{40 - 100}}{{20}} = \frac{{-60}}{{20}} = **-3.0** \] ### Summary of Standardized Scores: | Test Score (\( x \)) | Z-Score (\( z \)) | |----------------------|--------------------| | 120 | 1.0 | | 140 | 2.0 | | 80 | -1.0 | | 100 | 0.0 | | 40 | -3.0 | **Interpretation of Z-Scores:** - A **z-score** indicates how many standard deviations an element is from the mean. - Positive z-scores indicate scores **above** the mean. - Negative z-scores indicate scores **below** the mean. - A z-score of **0** indicates a score exactly at the mean. For example, a test score of **140** has a z-score of **2.0**, meaning it is **2 standard deviations above** the mean.