Question Video: Estimating Areas Under a Normal Distribution Curve | Nagwa Question Video: Estimating Areas Under a Normal Distribution Curve | Nagwa

Question Video: Estimating Areas Under a Normal Distribution Curve Mathematics • Third Year of Secondary School

For a normally distributed data set with mean 32.1 and standard deviation 2.8, between which two values would you expect 95% of the data set to lie?

03:14

Video Transcript

For a normally distributed data set with mean 32.1 and standard deviation 2.8, between which two values would you expect 95 percent of the data set to lie?

We recall firstly that for a normally distributed random variable, approximately 95 percent of the data points lie within two standard deviations of the mean. We therefore need to calculate the values two standard deviations below and two standard deviations above the mean for this particular normal distribution.

We’re given in the question that the mean is 32.1 and the standard deviation is 2.8, so we can calculate these values fairly easily. The lower value 𝜇 minus two 𝜎 is 32.1 minus two multiplied by 2.8, which is 26.5. The upper value 𝜇 plus two 𝜎 is 32.1 plus two times 2.8, which is 37.7. And so by recalling part of the empirical rule for a normally distributed random variable, which tells us that approximately 95 percent of the data set lies within two standard deviations of the mean, we find that for this distribution, 95 percent of the data set will lie between 26.5 and 37.7.

More generally, we may want to find the proportion of points that lie in other regions under the curve. To do this, we need to consider one special case of the normal distribution, which is what we call the standard normal distribution. We usually denote this using the letter 𝑧. And it represents the normal distribution which has a mean of zero and a standard deviation, and hence variance, of one.

Values from this distribution are known as 𝑧-scores, and they represent the number of standard deviations above the mean a particular value is. For example, a 𝑧-score of 1.4 would mean a value 1.4 standard deviations above the mean, whereas a 𝑧-score of negative 2.1 would mean a value 2.1 standard deviations below the mean. These 𝑧-scores for a standard normal distribution are really useful because they allow us to view values from a normal distribution on a standardized scale.

We have a set of statistical tables which we’ll look at in detail later, in which we can look up the areas and hence the probabilities associated with particular 𝑧-scores. The type of tables we’re going to use are tables which give the probability that our random variable capital 𝑍 is between zero and an observation lowercase 𝑧. That is the proportion of points or the area between zero and a positive 𝑧-score. If we wanted to then work out the proportion of points that lie completely to the left, that is, that are completely less than a particular positive 𝑧-score, we would need to add on 0.5 to the value from our tables to account for the area to the left of the axis of symmetry. That’s the area shaded in pink.

Join Nagwa Classes

Attend live sessions on Nagwa Classes to boost your learning with guidance and advice from an expert teacher!

  • Interactive Sessions
  • Chat & Messaging
  • Realistic Exam Questions

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy