Normal distribution, sampling, function fitting & Regression

Introduction

In this chapter, you will use the mean, median, mode and standard deviation of a set of data to identify whether the data is normally distributed or whether it is skewed. You will learn more about populations and selecting different kinds of samples in order to avoid bias. You will work with lines of best fit, and learn how to find a regression equation and a correlation coefficient. You will analyse these measures in order to draw conclusions and make predictions.

A normal distribution

Investigation :

You are given a table of data below.

1. Calculate the mean, median, mode and standard deviation of the data.
2. What percentage of the data is within one standard deviation of the mean?
3. Draw a histogram of the data using intervals $60\le x<64$ , $64\le x<68$ , etc.
4. Join the midpoints of the bars to form a frequency polygon.

If large numbers of data are collected from a population, the graph will often have a bell shape. If the data was, say, examination results, a few learners usually get very high marks, a few very low marks and most get a mark in the middle range. We say a distribution is normal if

• the mean, median and mode are equal.
• it is symmetric around the mean.
• $\pm 68\%$ of the sample lies within one standard deviation of the mean, $95\%$ within two standard deviations and $99\%$ within three standard deviations of the mean.

What happens if the test was very easy or very difficult? Then the distribution may not be symmetrical. If extremely high or extremely low scores are added to a distribution, then the mean tends to shift towards these scores and the curve becomes skewed.

If the test was very difficult, the mean score is shifted to the left. In this case, we say the distribution is positively skewed , or skewed right . If it was very easy, then many learners would get high scores, and the mean of the distribution would be shifted to the right. We say the distribution is negatively skewed , or skewed left .

Normal distribution

1. Given the pairs of normal curves below, sketch the graphs on the same set of axes and show any relation between them. An important point to remember is that the area beneath the curve corresponds to 100%.
   1. Mean = 8, standard deviation = 4 and Mean = 4, standard deviation = 8
   2. Mean = 8, standard deviation = 4 and Mean = 16, standard deviation = 4
   3. Mean = 8, standard deviation = 4 and Mean = 8, standard deviation = 8
2. After a class test, the following scores were recorded:

   Test Score	Frequency
   3	1
   4	7
   5	14
   6	21
   7	14
   8	6
   9	1
   Total	64
   Mean	6
   Standard Deviation	1,2

   1. Draw the histogram of the results.
   2. Join the midpoints of each bar and draw a frequency polygon.
   3. What mark must one obtain in order to be in the top 2% of the class?
   4. Approximately 84% of the pupils passed the test. What was the pass mark?
   5. Is the distribution normal or skewed?
3. In a road safety study, the speed of 175 cars was monitored along a specific stretch of highway in order to find out whether there existed any link between high speed and the large number of accidents along the route. A frequency table of the results is drawn up below.

   Speed (km.h ${}^{-1}$ )	Number of cars (Frequency)
   50	19
   60	28
   70	23
   80	56
   90	20
   100	16
   110	8
   120	5

   The mean speed was determined to be around 82 km.h ${}^{-1}$ while the median speed was worked out to be around 84,5 km.h ${}^{-1}$ .
   1. Draw a frequency polygon to visualise the data in the table above.
   2. Is this distribution symmetrical or skewed left or right? Give a reason fro your answer.