Ch. 3 – part 1•Measures of central tendency

•Measures of variation •Calculating standard deviation

Using the TI30XII

ma260notes_ch3_calc_directions.pptx

Notation- sample and populationsize Mean Variation 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, mode, midrange

Salary 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

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 = 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 formula calculation formula

S = =

  

 

Optional: Proof that the 2 s formulas are algebraically equivalent

Use theoretical formula to find st dev (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

Ex # 5 -- Computation formula

Use the computational formula for s

Xi xi 2

What does s mean?

For the previous example, calculate:

Mean + s = Mean – s =

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…

TI30XII continued…

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

8. 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.

Basic info on using the TI83 instead:

• 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: Use calculation formula for s

X237121516

Now, verify mean and s on your calculator

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.