Ch. 12 – part 1B•Measures of central tendency•Measures of variation •Shapes of distributions

•Calculation standard deviationUsing the TI30XII

Notation- sample and populationsize Mean Standard

deviation

Sample n s

Population N

Measures of Central Tendency-- Averages

• Find the average for the following test scores: Ex. #1: 78 83 97 32 75 45 52

How should we measure the average?

78 83 97 32 75 45 52

• Mean

• Median

• Mode

• Midrange

Ex. #2: Find mean, median, modeSalary Frequency

10,000 1

20,000 4

30,000 3

250,000 1

Ex #3: GPAs – calculateClass Grade # credits

Math B (3.0) 4

English C (2.0) 2

Physics A (4.0) 5

Skewed distributions and measures of central tendency

Ex. #4: Find the mean and median for each of the 2 examples

Class 1

100

90

70

50

40

Class 2

72

71

70

69

68

How do they vary?-- RangeClass 1

100

90

70

50

40

Class 2

72

71

70

69

68

Measures of variation

• Range=

• Sample Standard deviation (Theoretical)= s =

Standard deviation- Use theoretical formula to find s.

Then find mean + s and mean – s.

Class 1

100

90

70

50

40

Class 2

72

71

70

69

68

Example #5- Mean and s

Try an example using both the theoretical formula and the computational formula for s:

Data set: 1 1 2 4 7Calculate the mean.

You should get 3

Ex #5-- Theoretical formula

Use the theoretical formula for s

xi 2

A second formula- the computational formula for s

• Sometimes the theoretical calculations get messy, and we shouldn’t round in the middle of the problem. So we will need a computational formula which is algebraically equivalent. (It takes 5 steps of algebra).

S = =

theoretical computational

Ex # 5 -- Computation formula

Use the computational formula for s

Xi xi 2

3rd approach: Using your TI30XIIS for 1-Var Stats (using Ex #5)

Considering the following data set, calculate sample mean and sample standard deviation:

1 1 2 4 7 Here are the key strokes:1. Clear previous data with [EXIT STAT]: push [2nd] and then [STAT

VAR] (If you get an Error, hit CLEAR).2. Enter Statistics mode by hitting: the [2nd]button followed by [DATA]3. Hit [=] to accept One-Variable Mode.4. Hit the [DATA] button and it is ready for you to enter your first

value when it prompts X1=5. Type in the first piece of data (in this case it is 1).6. Hit the “down arrow” button to accept the piece of data.7. …

Directions continued7. When it say FRQ=1 (i.e. frequency is one) hit the “down arrow”

button again8. Now enter in your second piece of data when it prompts X2=9. Keep entering in data with frequencies of 1 until all of the data is

in the calculator. (In this case, after the 5 data values, the calculator prompts X6=).

10. Hit the [STATVAR] button.11. Use your right arrow to find the sample mean and sample

standard deviation (sx=2.55)12. Clear data again with [EXIT STAT]: Hitting [2nd] followed by

[STATVAR] will prompt you to leave the statistics mode. Hit [=] to leave statistics mode. You are now ready to start a new data set.

To use the TI83:

• Go to STAT/Edit: Pick 4. Type "ClrList L1"• Go to STAT/Edit Pick 1. Edit. Enter your list of

numbers.• Go to STAT/CALC and pick 1. 1-Var Stats

Ex #6: find st dev (s) with computational formula

6 6 6 6 7 7 7 4 8 3

Verify your work on a calculator

Ex. #7: use calculator to find mean, s:

78 73 84 72 66 42 9967 62 45 89 76 43 9856 78 66 54 88 67 90

• Also, find mean + smean - s

• Review stem/leaf

Ex. #8: Use the calculator to find s

73 58 68 75 94 87 83 89 52 99 95 77 75 81 75 69 76 77 71 50 Use the calculator to find s

Find mean + s= mean – s =