Download Use the computational formula for s X

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 population
size
Sample
n
Population
N
Mean
Variation
Standard
deviation
s
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 – calculate
Class
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
Symmetric
Skewed Left
Uniform
Skewed Right
Bimodal
Skewed distributions and measures of
central tendency
Ex. #4: Find the mean and median for
each of the 2 examples
Class 1
Class 2
100
72
90
71
70
70
50
69
40
68
How do they vary?-- Range
Class 1
Class 2
100
72
90
71
70
70
50
69
40
68
Measures of variation
• Range=
• Sample Standard deviation = s =
Basic formulas
size
Mean
Variance
Standard deviation
(algebraically
equivalent)
=
Population
=
N
theoretical
s2
Sample
n
=
s =
Computation-shortcut
s=
Sample Standard deviation (s) formulas–
Theoretical formula
S=
calculation formula
=
Optional: Proof that the 2 s formulas
are algebraically equivalent
Use theoretical formula to find st dev (s)
Class 1
Class 2
100
72
90
71
70
70
50
69
40
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
7
Calculate 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.
8.
9.
10.
11.
12.
When it say FRQ=1 (i.e.frequency is one) hit the “down
arrow” button again
Now enter in your second piece of data when it prompts
X2=
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=).
Hit the [STATVAR] button.
Use your right arrow to find the sample mean and
sample standard deviation (sx=2.55)
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
X
2
3
7
12
15
16
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
112.8
141.3
198.9
200.4
87.5
Sum=740.9
s=
xi 2
12723.84
19965.69
39561.21
40160.16
7656.25
sum=120,067.15
=
Now, verify your work using the TI30XII.
= 50.7