What is z-score and how is it used?
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Explanation and use of z-score.
Take an example if you want to make the interviewer understand the concept
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In finance, Z-scores are measures of an observation’s variability and can be used by traders to help determine market volatility. The Z-score is also sometimes known as the Altman Z-score. A Z-Score is a statistical measurement of a score’s relationship to the mean in a group of scores.For example, knowing that someone’s weight is 150 pounds might be good information, but if you want to compare it to the “average” person’s weight, looking at a vast table of data can be overwhelming (especially if some weights are recorded in kilograms). A z-score can tell you where that person’s weight is compared to the average population’s mean weight.
The sample mean’s distribution is assumed to be normal and z-score is calculated for a standardized normal distribution, standardized where Z=(x-u)/(o/sqrt(n)), where x is sample mean, u is population mean and o is sample standard deviation.
Right now, we do not know the value of u, hence Z-score is used as a test statistic for evaluating how confidently (usually 95%) we can ascertain that the population mean lies in a particular interval (calculated using a z-score, sample mean, sample size and population variance).
A Z score of 0 means, the observation is equal to mean.
A Z score of 1 means, the observation is 1 S.D above the mean.
Similarly a -ve Z score tells how below the observation is from mean.
z = (x – μ) / σ
Z-score means is a standardised score which tells us how many standard deviations above or below a data point is from the mean of the population. To know this, we only require the mean and standard deviation of the popoulation. Let’s understand this with the help of an example-
If we know your score in an exam( asume it to be 150 out of 200), you might wonder how you have performed as compared to other applicants. Here we use the Z-score = (your score- mean score)/Standard deviation.
Let this value be 1.96. Now using a normal table, we find the probability to be .975. This means that the probability that an applicant would have a score of 150 or less is 0.975. So, there are 97.5% people who scored below you.
z-score is calculated to standardized any value. It tells how many standard deviation the data value is above or below the mean.
It is one of the parameter to know how far the point is from mean and also to compare two points whose mean and standard deviation is different.
For example a boy score 80 marks in a class whose average is 70 and standard deviation is 10. The z-score of this boy is 1 whereas another boy who score 80 marks in another class whose average is 50 and standard deviation is 10 , the z score will be 3. This all imply that though the marks were same for both the boys but second boy is better as he score far above the average.
A Z-score is a standardized score that has a specific interpretation.
For instance, the mean rating of a product made available to 100 customers in the range of 1 to 10 is found to be 8.5 and the standard deviation is found to be 1.75
This is the mean and standard deviation of the population since the product has been made available to exactly 100 customers initially (maybe for testing or acceptance rate).
What is the probability of getting a rating of 6.5 if a random customer’s rating is checked from the population?
For this, we will have to standardize this value of 6.5 by subtracting it from the population mean and dividing the result by the population standard deviation. This will give us a Z-score which can then be used to find the probability. This probability is obtained based on the amount of area it covers under the normal distribution curve.
We will obtain a Z-score of -1.1429
A negative Z-score means that the data value is smaller than the mean and a positive Z-score means the data value is greater than the mean. It also tells how many standard deviations it is away from the population mean.
For the above-considered case, we can say that our sample data point (6.5) is approximately 1.14 standard deviations below the mean.