8 Data Analysis 2
8.3 The Normal Curve
Learning Objectives
After completing this section the student should be able to:
- Recognize the characteristics of a normal distribution
- Find scores at a designated standard deviation from the mean
- Interpret and use the 68-95-99.7 Rule
The Normal Distribution
The Galton Board, invented by Sir Francis Galton, consists of a vertical board with interleaved rows of pegs. Beads are dropped from the top and, when the device is level, bounce either left or right as they hit the pegs. Eventually they are collected into bins at the bottom, where the height of bead columns accumulated in the bins approximate a normal distribution. (https://en.wikipedia.org/wiki/Bean_machine#/)
In this section we will explore the normal distribution and the dispersion of data values around the mean. We have seen that the standard deviation provides a measure of the dispersion of the data values around the mean. If the standard deviation is zero then all data vales will equal the mean. The general idea seems to be that as the standard deviation increases the data will be more widely dispersed around the mean. We have also seen that data can be distributed in a variety of ways. Consider the histograms in Figures 1, 2 & 3. These histograms represent the evaluation scores (on a scale of 1 to 5) for three instructors. In all three cases a group of 10 students provided feedback for each of the instructors.
Referring to Figure 1, Instructor A received each possible score two times. Figure 1 represents a uniform distribution since every data value occurs with the same frequency.
Referring to Figure 2, Instructor B received a mix of scores. Figure 2 represents a skewed distribution where one tail of the distribution is stretched out more than the other.
Referring to Figure 3, Instructor C also received a mix of scores. Figure 3 represents a symmetrical distribution. Data values occur most often in the centre of the distribution and spread out equally on either side.
The histogram in Figure 3 is symmetric but because it represents a very small sample size it appears to be a series of rectangles stacked side by side. As the sample size increases a symmetrical distribution will become less boxy as illustrated in Figures 4 & 5.
Eventually if we consider the entire population the distribution will approach what is called the normal distribution as in Figure 6. This distribution is also called the bell curve.
The normal distribution models many aspects of real life, including height, blood pressure and IQ scores.
Normal Distribution
The normal distribution is also called the bell curve.
In a normal distribution the data values are symmetrical around a vertical line drawn through its centre which is also where the mean is located. Half of the data values lie on either side of the mean.
In a normal distribution the mean, median and mode will all be equal.
Normal Distribution and Standard Deviation
The normal distribution will have symmetry in rela