CHAPTER 11 Mean and Standard Deviation

BOX AND WHISKER PLOTS  Worksheet on Interpreting and making a box and whisker plot in the calculator

EXPECTED OUTCOMES  Understand the terms mean, median, mode, standard deviation  Use these terms to interpret data

MEASURES OF CENTRAL TENDENCY  Mean … the average score  Median … the value that lies in the middle after ranking all the scores  Mode … the most frequently occurring score

Which measure of Central Tendency should be used? MEASURES OF CENTRAL TENDENCY

The measure you choose should give you a good indication of the typical score in the sample or population.

MEASURES OF CENTRAL TENDENCY Mean … the most frequently used but is sensitive to extreme scores e.g Mean = 5.5 (median = 5.5) e.g Mean = 6.5 (median = 5.5) e.g Mean = 14.5 (median = 5.5)

MEASURES OF CENTRAL TENDENCY Median … is not sensitive to extreme scores … use it when you are unable to use the mean because of extreme scores

MEASURES OF CENTRAL TENDENCY Mode … does not involve any calculation or ordering of data … use it when you have categories (e.g. occupation)

THE NORMAL DISTRIBUTION CURVE In everyday life many variables such as height, weight, shoe size and exam marks all tend to be normally distributed, that is, they all tend to look like a bell curve.

Length of Right Foot Number of People with that Shoe Size Suppose we measured the right foot length of 30 teachers and graphed the results. Assume the first person had a 10 inch foot. We could create a bar graph and plot that person on the graph. If our second subject had a 9 inch foot, we would add her to the graph. As we continued to plot foot lengths, a pattern would begin to emerge. Your next mouse click will display a new screen.

Length of Right Foot If we were to connect the top of each bar, we would create a frequency polygon. Notice how there are more people (n=6) with a 10 inch right foot than any other length. Notice also how as the length becomes larger or smaller, there are fewer and fewer people with that measurement. This is a characteristics of many variables that we measure. There is a tendency to have most measurements in the middle, and fewer as we approach the high and low extremes. Number of People with that Shoe Size Your next mouse click will display a new screen

Length of Right Foot Number of People with that Shoe Size You will notice that if we smooth the lines, our data almost creates a bell shaped curve

Length of Right Foot Number of People with that Shoe Size You will notice that if we smooth the lines, our data almost creates a bell shaped curve. This bell shaped curve is known as the “Bell Curve” or the “Normal Curve.” Your next mouse click will display a new screen

Whenever you see a normal curve, you should imagine the bar graph within it Points on a Quiz Number of Students Your next mouse click will display a new screen.

The mean,mode,and median Points on a Quiz Number of Students = / 51 = , 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 22 will all fall on the same value in a normal distribution. Now lets look at quiz scores for 51 students. Your next mouse click will display a new screen.

THE NORMAL DISTRIBUTION CURVE It is bell-shaped and symmetrical about the mean The mean, median and mode are equal Mean, Median, Mode It is a function of the mean and the standard deviation

Normal distributions (bell shaped) are a family of distributions that have the same general shape. They are symmetric (the left side is an exact mirror of the right side) with scores more concentrated in the middle than in the tails. Examples of normal distributions are shown to the right. Notice that they differ in how spread out they are. The area under each curve is the same. Your next mouse click will display a new screen.

QUESTION: TWO CLASSES TOOK A RECENT QUIZ. THERE WERE 10 STUDENTS IN EACH CLASS, AND EACH CLASS HAD AN AVERAGE SCORE OF 81.5 SINCE THE AVERAGES ARE THE SAME, CAN WE ASSUME THAT THE STUDENTS IN BOTH CLASSES ALL DID PRETTY MUCH THE SAME ON THE EXAM?

THE ANSWER IS… NO. THE AVERAGE (MEAN) DOES NOT TELL US ANYTHING ABOUT THE DISTRIBUTION OR VARIATION IN THE GRADES.

Mean

SO, WE NEED TO COME UP WITH SOME WAY OF MEASURING NOT JUST THE AVERAGE, BUT ALSO THE SPREAD OF THE DISTRIBUTION OF OUR DATA.

MEASURES OF DISPERSION A measure of the central tendency by itself can be misleading Example: Two nations with the same median family income are very different if one has extremes of wealth and poverty and the other has little variation among families.

SO, WE NEED TO COME UP WITH SOME WAY OF MEASURING NOT JUST THE AVERAGE, BUT ALSO THE SPREAD OF THE DISTRIBUTION OF OUR DATA.

WHY STUDY DISPERSION?  A small value for a measure of dispersion indicates that the data are clustered closely (the mean is therefore representative of the data)  A large measure of dispersion indicates that the mean is not reliable (it is not representative of the data)

DISPERSION Dispersion refers to how much the data are spread out. Analogy: In terms of physical fitness, a person which can do the “splits” is more agile than one who can not. The agile one can spread out more! Data sets that are more disperse are spread out more. Other names for dispersion are variability, variation or spread. So, when we look at a variable we can look at the variation. This is the amount of scattering of the values away from the central value.

27 DEFINITION  Measures of dispersion (Variation) describe how similar a set of scores are to each other  The more similar the scores are to each other, the lower the measure of dispersion will be  The less similar the scores are to each other, the higher the measure of dispersion will be  In general, the more spread out a distribution is, the larger the measure of dispersion will be

28 MEASURES OF DISPERSION  Which of the distributions of scores has the larger dispersion? The upper distribution has more dispersion because the scores are more spread out That is, they are less similar to each other

VARIATION OR SPREAD OF DISTRIBUTIONS Measures that indicate the spread of scores:  Range  Standard Deviation

VARIATION OR SPREAD OF DISTRIBUTIONS Range  It compares the minimum score with the maximum score  Max score – Min score = Range  It is a crude indication of the spread of the scores because it does not tell us much about the shape of the distribution and how much the scores vary from the mean

WELL, FOR EXAMPLE, LETS SAY FROM A SET OF DATA, THE AVERAGE IS AND THE RANGE IS 23. But what if the data looked like this:

Here is the average And here is the range But really, most of the numbers are in this area, and are not evenly distributed throughout the range.

VARIATION OR SPREAD OF DISTRIBUTIONS Standard Deviation  It tells us what is happening between the minimum and maximum scores  It tells us how much the scores in the data set vary around the mean  It is useful when we need to compare groups using the same scale

STANDARD DEVIATION Standard Deviation shows the variation in data. It is a measurement that tells how on average each data point, x, deviates from the mean. -If the data is close together, the standard deviation will be small. If the data is spread out, the standard deviation will be large. Standard Deviation is often denoted by the lowercase Greek letter sigma,.

The mean and standard deviation are useful ways to describe a set of scores. If the scores are grouped closely together, they will have a smaller standard deviation than if they are spread farther apart. Small Standard Deviation Large Standard Deviation Click the mouse to view a variety of pairs of normal distributions below. Different Means Different Standard Deviations Different Means Same Standard Deviations Same Means Different Standard Deviations Your next mouse click will display a new screen.

NOW, LETS COMPARE THE TWO CLASSES AGAIN Team ATeam B Average on the Quiz Standard Deviation

The following table shows the results of student A and student B in a test. Which student shows a more consistent performance? Why? Student Mean marks Standard Deviation AB

ANSWER The performance of student B is more consistent because the standard deviation of his marks is smaller.

The table below shows the marks for a test of 3 classes in form 5. (a) Between 5A and 5C, which class show a more consistent achievement? Class Mean marks Standard Deviation Numbers of students 5A5B5C

(a) Class 5A shows a more consistent achievement because the standard deviation of its marks is smaller.

STANDARD DEVIATION Deviation just means how far from the normal

POPULATION STANDARD DEVIATION  The population standard deviation uses the squares of the residuals  Steps;  Find the sum of the squares of the residuals  Find the mean  Then take the square root of the mean  Formula:

STANDARD DEVIATION  So if that’s what it is, how do you find it???  The formula is easy: it is the square root of the Variance. So now you ask, "What is the Variance?"

VARIANCE  The average of the squared differences from the Mean.  To calculate the variance follow these steps:  Work out the Mean (the simple average of the numbers)Mean  Then for each number: subtract the Mean and then square the result (the squared difference).  Then work out the average of those squared differences. (Why Square?)Why Square?

EXAMPLE  You and your friends have just measured the heights of your dogs (in millimeters):  The heights (at the shoulders) are: 600mm, 470mm, 170mm, 430mm and 300mm.  Find out the Mean, the Variance, and the Standard Deviation.

FIND THE MEAN Mean = = 1970 =

FIND THE VARIANCE (MEAN OF THE DIFFERENCES)  Now, we calculate each dogs difference from the Mean: (mean was 394) Variance: σ 2 = (-224) (-94) 2 = 108,520 = 21, To calculate the Variance, take each difference, square it, and then average the result: So, the Variance is 21,704.

*NOTE: WHY SQUARE ?  Squaring each difference makes them all positive numbers (to avoid negatives reducing the Variance)  And it also makes the bigger differences stand out. For example =10,000 is a lot bigger than 50 2 =2,500.  But squaring them makes the final answer really big, and so un- squaring the Variance (by taking the square root) makes the Standard Deviation a much more useful number.

YOU TRY What is the standard deviation for the numbers: 75, 83, 96, 100, 121 and 125? (Try to do this yourself, without using the Standard Deviation calculator.) (Here is your chance to practice your skills.) A 16.9 B 17.1 Question C 17.6 D 18.2

1. Firstly find the mean: Mean = ( ) ÷ 6 = 600 ÷ 6 = Next find the variance. To calculate the Variance, take each difference, square it, and then average the result: ( ) 2 + ( ) 2 + ( ) 2 + ( ) 2 + ( ) 2 + ( ) 2 = (-25) 2 + (-17) 2 + (-4) 2 + (0) 2 + (21) 2 + (25) 2 = = 1,996 So the Variance = 1,996 ÷ 6 = The Standard Deviation is just the square root of the Variance = √( ) = 18.2 correct to 2 decimal places How it’s done!

CAN WE FIND THE STANDARD DEVIATION WITHOUT ALL THE WORK? Lets take the same data we just used What is the standard deviation for the numbers: 75, 83, 96, 100, 121 and 125? Our answer was 18.2 But now lets use the calculator to make this task a bit less tedious

MEAN AND STANDARD DEVIATION  Finish packet with Box and whisker on front.

Using Standard Deviation 23/english/academy123_content/wl-book- demo/ph-681s.html /Histogram/ Interactive Histogram site Interpreting Statistics site s/mathematics/mc2/cim/interactive_labs/M2_ 02/M2_02_dev_100.html Interactive Line Plot site