Page 183 - Data Science class 10
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Thus, the z-score is equal to the raw score minus the population mean, divided by the population standard
deviation. When we come across conditions where the population mean and the population standard deviation
are not known, the standard score can be determined using the sample mean, i.e., x ̄ and the sample standard
deviation as estimates of population values.
Now let us consider an example that will illustrate the use of the z-score formula. Suppose that we know about a
population of kids with weights that are normally distributed. Further to this, consider that we know that the mean
of the distribution is 12 kg and the standard deviation is 2 kg. Now consider the following questions:
1. What is the z-score for 15 kgs?
2. What is the z-score for 8 kgs?
3. How many kgs correspond to a z-score of 2.25?
1. Here, x = 15
μ = 12
σ = 2
(x – μ)
z =
σ
= 15 – 12
2
= 1.5
This means that 15 is 1.5 standard deviations above the mean.
2. Here, x = 8
μ = 12
σ = 2
(x – μ)
z =
σ
= 8 – 12
2
= – 2
This means that 8 is 2 standard deviations below the mean.
3. Here, z = 2.25
Put z = 2.25 into the formula and use basic algebra to solve for x:
2.25 = (x – 12)
2
Multiply both the sides by 2:
4.5 = (x – 12)
Add 12 to both the sides:
16.5 = x
Hence, we see that 16.5 kgs correspond to a z-score of 2.25.
Let us take an another example of z-score.
Example: Rohan scored 80 marks in a test. The mean score of the class was 50 with a standard deviation of 10.
Calculate the z score for the marks secured by Rohan using the z score formula.
To find: z score for marks secured by Rohan
Given:
Marks secured by Rohan, x = 80
Data Merging 181

