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Normal Distribution Homework Help

Also known as the Bell curve or Gaussian distribution, a normal distribution is a probability function used to show how values of a given variable are distributed. It occurs naturally in many different situations like in people’s heights, sizes of items produced by a machine, marks on a test, salaries, IQ scores, errors in measurements, etc. In a normal distribution, most of the data values cluster around the central peak (mean), and the longer the distance between the value and the mean, the less likely the event is to occur.

Properties of a normal distribution

There are a few things that differentiate a normal distribution from other distributions. For instance:

  • The mean, median, and mode are all equal
  • The curve is symmetrical around the mean (at the center)
  • Half of the values displayed on the curve are to the right of the mean and another half is to the left
  • The area under the curve is always one

In addition to the above properties, a normal distribution curve contains skewness and kurtosis, coefficients used to measure the symmetry of the distribution, and the thickness of the ends of the tails respectively.

Parameters of normal distribution

A normal distribution has two main parameters: the mean and standard deviation. These two parameters determine the probabilities and shape of the distribution. When changes are applied to the parameter values, the shape of the distribution also changes.


The mean is a measure of central tendency, used to explain the distribution of variables that have been measured as intervals or ratios. When plotting normal distribution graphs, the mean is used to define the peak location and the majority of data points are distributed around the peak. When a change is applied to the mean values, the curve shifts either right or left along the x-axis.

Standard deviation:

The standard deviation is used to measure how data points are dispersed in relation to the mean. It shows the exact distance between the data points and the mean and how these data points are positioned. On a normal distribution graph, the standard deviation measures the curve width, and changes to data on the graph either expands or tightens the distribution width along the x-axis. Typically, when there is a slight standard deviation in relation to the mean, the curve becomes steep and when there is a larger standard deviation, the curve becomes flatter.

Practical applications of normal distribution models

The normal distribution can help you find out which subjects you are good at and which ones you need to put in a little more work. When you score a higher grade in one subject and score poorer in another, you will certainly think that you scored higher in this one subject because you are better at it, but that’s not always true. The only time you can say that you are good at a certain subject is if the mark you get has a specific number of standard deviations over, or rather, above the mean. As mentioned earlier, standard deviation shows you how data is distributed around the mean. It enables you to make comparisons between different distributions including different data types and means. If you score 95 in Chemistry and 90 in Math, for instance, you may think you are good at Chemistry than you are in Math. But in Chemistry, the score is one standard deviation over the mean, while in Math, it is two standard deviations over the mean. This shows that your score in Math is way higher than what most students scored. Looking at the results of the standard deviation you actually scored better in Chemistry than in Math.