Question

A normal distribution curve, where x = 70 and σ = 15, was created by a teacher using her students’ grades. What information about their performances can be obtained by analyzing the curve?

Answer 1

sample answer: By analyzing the curve, you can tell that the average, or mean, grade is 70. This is also the median of the grades, so we know that one half of the scores are less than or equal to 70, and the other half of the scores are greater than or equal to 70. The bulk of the scores are between 55 and 85.

Answer 2

The median and mode of the students grade is 70.

Most of the students scored between 40 and 100.

Explanation:

From the provided information it can be seen that the mean of the distribution is, μ = 70 and the standard deviation is, σ = 15.

For a Normal distributed data the mean, median and mode are the same.

So, the median and mode of the students grade is 70.

The standard deviation of the data represents the spread of the observation, i.e. how dispersed the values are along the curve.

In statistics, the 68–95–99.7 rule, also recognized as the empirical rule, is a shortcut used to recall that 68.27%, 95.45% and 99.73% of the values of a Normally distributed data lie within one, two and three standard deviations of the mean, respectively.

`P(\mu-\sigma<X<\mu+\sigma)=P(70-15<X<70+15)=P(55<X<85)=0.68`

`P(\mu-2\sigma<X<\mu+2\sigma)=P(70-30<X<70+30)=P(40<X<100)=0.95`

`P(\mu-3\sigma<X<\mu+3\sigma)=P(70-45<X<70+45)=P(25<X<115)=0.997`

Assuming that maximum marks of the exam is 100, it can be said that most of the students scored between 40 and 100.