Z score : Definition & Interpretation
Sometimes researcher asks a question as a specific observation is common or exceptional. It can be answered as no. of standard deviations removed from mean. It’s called Z score. Z score gives the information about how extreme an observation is.
It is useful to standardized the values of a normal distribution by converting them into z-scores because:
(a) It allows to calculate the probability of a score occurring within a standard normal distribution;
(b) It enables us to compare two scores that are from different samples (may have different means and standard deviations).
Z score calculation
The formula for calculating a Z-score is is z = (x-μ)/σ, where x is the raw score, μ is the population mean, and σ is the population standard deviation.
How can we decide if a certain Z score is high or low. It depends on context and distribution.
- If it is a normal distribution
In this case we can say z score >3 or z score<-3 can be assumed as rather exceptional.
2. If it is skewed to right.
Large positive Z scores are more common in right skewed graph.
3. If it is skewed to left.
Large negative Z scores are more common as more extreme values are on left side of distribution.
For any distribution, regardless shape it is said that 75% of data must lie between -2 and + 2 Z score and 89% of data between -3 and 3 Z score values.
Z score vs t score
There are few topics in statistics which cause confusion for beginners as when we should use Z score and when we should go for t score. They both are used in hypothesis testing.
Difference
- Z score is preferred for standardisation from the population raw data or more than 30 sample data to a standard score while t score is standardisation from the sample data of less than 30 data to standard score.
- For example, Z score is extensively used in stock market data and to check chances of a company going into bankrupt, while t score is extensively used in checking bone mineral density and Fracture risk assessments
This flow chart explain selection of Z score vs t score
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