Which of the following statements are correct for standard scores?
A. Mean of standard scores is 0.
B. Standard deviation of standard scores is 1.
C. Standard scores is free from units.
D. Limits of standard scores are ±2
E. Standard scores are always positive.
Choose the correct answer from the options given below:

Correct option is B

Standard scores (often known as z-scores) are used to express a value's relationship to the mean in terms of standard deviations. Let's examine each statement:
· A. Mean of standard scores is 0: This statement is correct. Standard scores are calculated as z=X−μσz = \frac{X - \mu}{\sigma}z=σX−μ​, where XXX is the individual data point, μ\muμ is the mean of the distribution, and σ\sigmaσ is the standard deviation. After standardizing, the mean of the transformed distribution becomes 0.
· B. Standard deviation of standard scores is 1: This statement is correct. The standard deviation of the distribution of standard scores (z-scores) is always 1, regardless of the original data distribution.
· C. Standard scores are free from units: This statement is correct. Standard scores are dimensionless because they represent the number of standard deviations a data point is from the mean, so they are independent of the original measurement units.
· D. Limits of standard scores are ±2: This statement is incorrect. There are no fixed limits for standard scores. A z-score can theoretically range from negative infinity to positive infinity, depending on how far a data point is from the mean. Typically, values within ±2 standard deviations (z-scores) represent the majority of the data (95% in a normal distribution), but the limits are not strictly ±2.
· E. Standard scores are always positive: This statement is incorrect. Standard scores can be positive or negative. Positive z-scores represent values above the mean, and negative z-scores represent values below the mean.
Thus, the correct statements are A, B, and C.
Information Booster:
1. Standard scores are used to compare scores from different distributions or scales, making them a valuable tool for normalization in statistics.
2. A z-score of 0 indicates that the value is exactly at the mean of the distribution.
3. The standard deviation of 1 means that the distribution of z-scores is measured in standard deviations, making it easier to compare scores from different data sets.
4. The idea of units being removed from standard scores is key in statistical analysis, especially when comparing datasets that use different measurement units.
Additional Information:
· Z-scores are also used to identify outliers in a dataset. A z-score greater than 3 or less than -3 generally indicates an outlier.
· The limits of z-scores vary depending on the data distribution, but standard scores themselves do not have inherent limits.