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)