Ch. 3 – part 1•Measures of central tendency

•Measures of variation •Calculation standard deviation

Using the TI30XII

Notation- sample and populationsize Mean Variation Standard

deviationProportion

Sample n s

Population N p

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 and medianSalary 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

Find the approximate mean, median, mode on these distributions

SymmetricUniform Bimodal

Skewed Left Skewed Right

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 = s =

Basic formulassize Mean Variance Standard deviation

(algebraically equivalent)

Population

N

=

 

=

    Sample

    n

  =

  s2

  theoretical s =

Computation-shortcut

s =

 

Sample Standard deviation (s) formulas– theoretical and shortcut

  

 

The 2 s formulas are algebraically equivalent

Standard deviation- theoretical formula

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 #4-- Theoretical formula

Use the theoretical formula for s

xi 2

Ex # 4 -- Computation formula

Use the computational formula for s

Xi xi 2

Using your TI30XIIS or TI30XIIB for One variable statistics (using Ex #4)

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. 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

Example #6: 112.8 141.3 198.9 200.4 87.5

When the mean isn’t an integer, the theoretical formula is messier. Try another example using only the computational formula for s. This is the one we’ll usually use:

Data set: 112.8 141.3 198.9 200.4 87.5 Calculate the mean. You should get 148.18

Calculate s, using shortcut formula xi xi

2

112.8 12723.84 141.3 19965.69 198.9 39561.21 200.4 40160.16 87.5 7656.25 Sum=740.9 sum=120,067.15 s = = = 50.7

Now, verify your work using the TI30XII.