1. Measures of Central Tendency

2. Objective
To learn how to find measures of central tendency in a set of raw data.

3. Relevance
To be able to calculate the most appropriate measure of center after analyzing the context of a study that might or might not contain extreme values.

4. There are 4 values that are considered measures of the center. 1. Mean
Central Values – Many times one number is used to describe the entire sample or population. Such a number is called an average. There are many ways to compute an average.
There are 4 values that are considered measures of the center.
1. Mean
2. Median
3. Mode
4. Midrange

5. Arrays
Mean – the arithmetic average with which you are the most familiar.
Mean:

6. Sample and Population Symbols
As we progress in this course there will be different symbols that represent the same thing. The only difference is that one comes from a sample and one comes from a population.

7. Symbols for Mean
Sample Mean:
Population Mean:

8. Rounding Rule
Round answers to one decimal place more than the number of decimal places in the original data.
Example: 2, 3, 4, 5, 6, 8
A Sample answer would be 4.1

9. Example
Find the mean of the array.
4, 3, 8, 9, 1, 7, 12

10. Example…….
Find the mean of the following numbers.
23, 25, 26, 29, 39, 42, 50

11. Example 2 – Use GDC
Find the mean of the array. 2.0, 4.9, 6.5, 2.1, 5.1, 3.2, 16.6 Use your lists on the calculator and follow the steps.

12. Stat, Edit – input list

13. Stat, Calc, One-Var Stats, L1

14. Or…..(I like this way better!)
Home: 2nd Stat Math 3: Mean (L#)

15. Rounding
The mean (x-bar) is 5.77.
We used 2 decimal places because our original data had 1 decimal place.

16. Median
Median – the middle number in an ordered set of numbers. Divides the data into two equal parts.
Odd # in set: falls exactly on the middle number.
Even # in set: falls in between the two middle values in the set; find the average of the two middle values.

17. Example
Find the median.
A. 2, 3, 4, 7, the median is 4.
B. 6, 7, 8, 9, 9, 10
median = (8+9)/ = 8.5.

18. Ex 2 – Use Calculator
Input data into L1.

19. Run “Stat, Calc, One-Variable Stats, L1”
Cursor all the way down to find “med”

20. Or…….from the home screen
2nd
Stat
Math
4: Median(L#)

21. Mode The number that occurs most often.
Suggestion: Sort the numbers in L1 to make it easier to see the grouping of the numbers.
You can have a single number for the mode, no mode, or more than one number.

22. Example
Find the mode.
1, 2, 1, 2, 2, 2, 1, 3, 3
Put numbers in L1 and sort to see the groupings easier.

23. The mode is 2.

24. Ex 2
Find the mode.
A. 0, 1, 2, 3, no mode
B. 4, 4, 6, 7, 8, 9, 6, ,6, and 9

25. Midrange
The number exactly midway between the lowest value and highest value of the data set. It is found by averaging the low and high numbers.

26. Example
Find the midrange of the set.
3, 3, 5, 6, 8

27. Trimmed Mean

28. Trimmed Mean
We have seen 4 different averages: the mean, median, mode, and midrange. For later work, the mean is the most important.
However, a disadvantage of the mean is that it can be affected by extremely high or low values.
One way to make the mean more resistant to exceptional values and still sensitive to specific data values is to do a trimmed mean.
Usually a 5% trimmed mean is used.

29. How to Compute a 5% Trimmed Mean
Order the data from smallest to largest.
Delete the bottom 5% of the data and the top 5% of the data. (NOTE: If 5% is a decimal round to the nearest integer)
Compute the mean of the remaining 90%.

30. Example a) Compute the mean for the entire sample.
b) Compute a 5% trimmed mean.

31. Example c) Compute the median for the entire sample.
d) Compute a 5% trimmed median.
The median is still 32.5.
e) Is the trimmed mean or the original mean closer to the median?
Trimmed Mean

32. Weighted Average

33. Sometimes we wish to average numbers, but we want to assign more importance, or weight, to some of the numbers.
The average you need is the weighted average.

34. Formula for Weighted Average

35. Example: You will earn an A! Suppose your midterm test score is 83
and your final exam score is 95.
Using weights of 40% for the midterm
and 60% for the final exam, compute
The weighted average of your scores.
If the minimum average for an A is
90, will you earn an A?
You will earn
an A!

36. Measures of Dispersion…..Arrays

37. Dispersion
The measure of the spread or variability
No Variability – No Dispersion

38. Measures of Variation
There are 3 values used to measure the amount of dispersion or variation. (The spread of the group)
1. Range
2. Variance
3. Standard Deviation

39. Why is it Important?
You want to choose the best brand of paint for your house. You are interested in how long the paint lasts before it fades and you must repaint. The choices are narrowed down to 2 different paints. The results are shown in the chart. Which paint would you choose?

40. The chart indicates the number of months a paint lasts before fading.
Paint A
Paint B 10 35 60 45 50 30 40 20 25
210

41. Does the Average Help?
Paint A: Avg = 210/6 = 35 months
Paint B: Avg = 210/6 = 35 months
They both last 35 months before fading. No help in deciding which to buy.

42. Consider the Spread Paint A: Spread = 60 – 10 = 50 months
Paint B: Spread = 45 – 25 = 20 months
Paint B has a smaller variance which means that it performs more consistently. Choose paint B.

43. Range
The range is the difference between the lowest value in the set and the highest value in the set.
Range = High # - Low #

44. Example
Find the range of the data set.
40, 30, 15, 2, 100, 37, 24, 99
Range = 100 – 2 = 98

45. Deviation from the Mean
A deviation from the mean, x – x bar, is the difference between the value of x and the mean x bar.
We base our formulas for variance and standard deviation on the amount that they deviate from the mean.
We’ll use a shortcut formula – not in book.

46. Variance (Array)
Variance Formula

47. Standard Deviation
The standard deviation is the square root of the variance.

48. Example – Using Formula
Find the variance.
6, 3, 8, 5, 3

50. Find the standard deviation
The standard deviation is the square root of the variance.

51. Same Example – Use Calculator
Put numbers in L1.

52. Run “Stat, Calc, One-Variable Stats, L1” and read the numbers
Run “Stat, Calc, One-Variable Stats, L1” and read the numbers. Remember you have to square the standard deviation to get variance.
Square this number
to get the variance!

53. Or….from the home screen
2nd Stat Math 7:stdDev(L1) Enter

54. Variance – Using Formula
Square the ENTIRE number for the standard deviation not the rounded version you gave for your answer.

55. Variance using GDC
2nd Stat Math 8: Variance (L1)