Z-Score Definition: The Z-Score also referred to as standardized raw scores (or simply standard score) is a useful statistic because not only permits to computation of the probability (chances or likelihood) of the raw score (occurring within normal distribution) but also helps to compare two raw scores from different normal distributions. The Z score is a dimensionless measure since it is derived by subtracting the population mean from an individual raw score and then this difference is divided by the population standard deviation. This computational procedure is called standardizing raw score, which is often used in the Z-test of testing of hypothesis.
Any raw score can be converted to a Z-score formula by
$$Z-Score=\frac{raw score – mean}{\sigma}$$
Z-Score Real Life Examples
Example 1: If the mean = 100 and standard deviation = 10, what would be the Z-score of the following raw score
| Raw Score | Z Scores |
|---|---|
| 90 | $ \frac{90-100}{10}=-1$ |
| 110 | $ \frac{110-100}{10}=1$ |
| 70 | $ \frac{70-100}{10}=-3$ |
| 100 | $ \frac{100-100}{10}=0$ |
Note that: If Z-Score,
- has a zero value then it means that the raw score is equal to the population mean.
- has a positive value then it means that the raw score is above the population mean.
- has a negative value then it means that the raw score is below the population mean.
Example 2: Suppose you got 80 marks in an Exam of a class and 70 marks in another exam of that class. You are interested in finding that in which exam you have performed better. Also, suppose that the mean and standard deviation of exam-1 are 90 and 10 and in exam-2 60 and 5 respectively. Converting both exam marks (raw scores) into the standard score, we get
$Z_1=\frac{80-90}{10} = -1$
The Z-score results ($Z_1=-1$) show that 80 marks are one standard deviation below the class mean.
$Z_2=\frac{70-60}{5}=2$
The Z-score results ($Z_2=2$) show that 70 marks are two standard deviations above the mean.
From $Z_1$ and $Z_2$ means that in the second exam, students performed well as compared to the first exam. Another way to interpret the Z score of $-1$ is that about 34.13% of the students got marks below the class average. Similarly, the Z Score of 2 implies that 47.42% of the students got marks above the class average.
Application of Z Score
- Identifying Outliers: The standard score can help in identifying the outliers in a dataset. By looking for data points with very high negative or positive z-scores, one can easily flag potential outliers that might warrant further investigation.
- Comparing Data Points from Different Datasets: Z-scores allow us to compare data points from different datasets because these scores are expressed in standard deviation units.
- Standardizing Data for Statistical Tests: Some statistical tests require normally distributed data. The Zscore can be used to standardize data (transforming it to have a mean of 0 and a standard deviation of 1), making it suitable for such tests.
Limitation of ZScores
- Assumes Normality: The Zscores are most interpretable when the data is normally distributed (a bell-shaped curve). If the data is significantly skewed, the scores might be less informative.
- Sensitive to Outliers: The presence of extreme outliers can significantly impact the calculation of the mean and standard deviation, which in turn, affects the standard score of all data points.
In conclusion, z-scores are a valuable tool for understanding the relative position of a data point within its dataset. The standard score offers a standardized way to compare data points, identify outliers, and prepare data for statistical analysis. However, it is important to consider the assumptions of normality and the potential influence of outliers when interpreting the z-scores.