Measures of Central Tendency Section 2.3. Central Values There are 4 values that are considered...
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Transcript of Measures of Central Tendency Section 2.3. Central Values There are 4 values that are considered...
Measures of Central Tendency
Section 2.3
Central Values
There are 4 values that are considered measures of the center.
1. Mean 2. Median 3. Mode 4. Midrange
Arrays
Mean – the arithmetic average with which you are the most familiar.
Mean: xofnumber
xallofsumbarx
n
xx
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.
Symbols for Mean
Sample Mean:
Population Mean:
x
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
Example
Find the mean of the array.
4, 3, 8, 9, 1, 7, 12
3.629.67
44
7
12719834
n
xx
Example 2 – Use Calculator
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.
Stat, Edit – input list
Stat, Calc, One-Var Stats, L1
Or…..(I like this way better!)
2nd Stat Math 3: Mean (L#)
Rounding
The mean (x-bar) is 5.77.
We used 2 decimal places because our original data had 1 decimal place.
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.
Example
Find the median.
A. 2, 3, 4, 7, 8 - the median is 4.
B. 6, 7, 8, 9, 9, 10
median = (8+9)/2 = 8.5.
Ex 2 – Use Calculator
Input data into L1.
Run “Stat, Calc, One-Variable Stats, L1”
Cursor all the way down to find “med”
Or…….
2nd
Stat Math 4: Median(L#)
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.
Example
Find the mode. 1, 2, 1, 2, 2, 2, 1, 3, 3 Put numbers in L1 and sort to see the
groupings easier.
The mode is 2.
Ex 2
Find the mode.
A. 0, 1, 2, 3, 4 - no mode
B. 4, 4, 6, 7, 8, 9, 6, 9 - 4 ,6, and 9
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.
2
ValueHighvalueLowmidrange
Example
Find the midrange of the set.
3, 3, 5, 6, 8
5.52
11
2
)83(
midrange
Measures of Dispersion…..Arrays
Section 2.4
Dispersion
The measure of the spread or variability
No Variability – No Dispersion
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
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 below. Which paint would you choose?
The chart indicates the number of months a paint lasts before fading.
Paint A Paint B
10 35
60 45
50 30
30 35
40 40
20 25
210 210
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.
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.
Range
The range is the difference between the lowest value in the set and the highest value in the set.
Range = High # - Low #
Example
Find the range of the data set.
40, 30, 15, 2, 100, 37, 24, 99
Range = 100 – 2 = 98
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.
Variance (Array)
Variance Formula
1
)( 22
2
n
n
xx
s
Standard Deviation
The standard deviation is the square root of the variance.
2ss
Example – By Hand
Find the variance.
6, 3, 8, 5, 3
6 363 98 645 253 9
x 2x
25 x 1432 x
5.44
18
4
125143
45
25143
2
2
s
1
)( 22
2
n
n
xx
s
Find the standard deviation
The standard deviation is the square root of the variance.
12.25.4 s
Same Example – Use Calculator
Put numbers in L1.
Run “Stat, Calc, One-Variable Stats, L1” and read the numbers. Remember you have to square the standard deviation to get variance.
Or….
2nd Stat Math 7:stdDev(L1) Enter
Variance – By Hand
Square the ENTIRE number for the standard deviation not the rounded version you gave for your answer.
5.4)121320344.2( 22 s
Variance on Calculator
2nd Stat Math 8: Variance (L1)