A z-score

A z-score, in simple terms, is a score that expresses the value of a distribution in standard deviation with respect to the mean. Let's take a look at the following formula that calculates the z-score:

Here, X is the value in the distribution, μ is the mean of the distribution, and σ is the standard deviation of the distribution

Let's try to understand this concept from the perspective of a school classroom.

A classroom has 60 students in it and they have just got their mathematics examination score. We simulate the score of these 60 students with a normal distribution using the following command:

>>> classscore
>>> classscore = np.random.normal(50, 10, 60).round()

[ 56. 52. 60. 65. 39. 49. 41. 51. 48. 52. 47. 41. 60. 54. 41.
 46. 37. 50. 50. 55. 47. 53. 38. 42. 42. 57. 40. 45. 35. 39.
 67. 56. 35. 45. 47. 52. 48. 53. 53. 50. 61. 60. 57. 53. 56.
 68. 43. 35. 45. 42. 33. 43. 49. 54. 45. 54. 48. 55. 56. 30.]

The NumPy package has a random module that has a normal function, where 50 is given as the mean of the distribution, 10 is the standard deviation of the distribution, and 60 is the number of values to be generated. You can plot the normal distribution with the following commands:

>>> plt.hist(classscore, 30, normed=True) #Number of breaks is 30
>>> plt.show()

The score of each student can be converted to a z-score using the following functions:

>>> stats.zscore(classscore)

[ 0.86008868 0.38555699 1.33462036 1.92778497 -1.15667098 0.02965823
 -0.91940514 0.26692407 -0.08897469 0.38555699 -0.20760761 -0.91940514
 1.33462036 0.62282284 -0.91940514 -0.32624053 -1.39393683 0.14829115
 0.14829115 0.74145576 -0.20760761 0.50418992 -1.2753039 -0.80077222
 -0.80077222 0.9787216 -1.03803806 -0.44487345 -1.63120267 -1.15667098
 2.16505081 0.86008868 -1.63120267 -0.44487345 -0.20760761 0.38555699
 -0.08897469 0.50418992 0.50418992 0.14829115 1.45325329 1.33462036
 0.9787216 0.50418992 0.86008868 2.28368373 -0.6821393 -1.63120267
 -0.44487345 -0.80077222 -1.86846851 -0.6821393 0.02965823 0.62282284
 -0.44487345 0.62282284 -0.08897469 0.74145576 0.86008868 -2.22436727]

So, a student with a score of 60 out of 100 has a z-score of 1.334. To make more sense of the z-score, we'll use the standard normal table.

This table helps in determining the probability of a score.

We would like to know what the probability of getting a score above 60 would be.

The standard normal table can help us in determining the probability of the occurrence of the score, but we do not have to perform the cumbersome task of finding the value by looking through the table and finding the probability. This task is made simple by the cdf function, which is the cumulative distribution function:

>>> prob = 1 - stats.norm.cdf(1.334)
>>> prob

0.091101928265359899

The cdf function gives the probability of getting values up to the z-score of 1.334, and doing a minus one of it will give us the probability of getting a z-score, which is above it. In other words, 0.09 is the probability of getting marks above 60.

Let's ask another question, "how many students made it to the top 20% of the class?"

Here, we'll have to work backwards to determine the marks at which all the students above it are in the top 20% of the class:

Now, to get the z-score at which the top 20% score marks, we can use the ppf function in SciPy:

>>> stats.norm.ppf(0.80)

0.8416212335729143

The z-score for the preceding output that determines whether the top 20% marks are at 0.84 is as follows:

>>> (0.84 * classscore.std()) + classscore.mean()

55.942594176524267

We multiply the z-score with the standard deviation and then add the result with the mean of the distribution. This helps in converting the z-score to a value in the distribution. The 55.83 marks means that students who have marks more than this are in the top 20% of the distribution.

The z-score is an essential concept in statistics, which is widely used. Now you can understand that it is basically used in standardizing any distribution so that it can be compared or inferences can be derived from it.