Lecture 17Sections 5.1 – 5.2Wed, Sep 19, 2007

Measuring Center

Measuring the Center Often, we would like to have one number

that that is “representative” of a population or sample.

It seems reasonable to choose a number that is near the “center” of the distribution rather than in the left or right extremes.

But there is no single “correct” way to do this.

Measuring the Center

Mean – the simple average of a set of numbers.

Median – the value that divides the set of numbers into a lower half and an upper half.

Mode – the most frequently occurring value in the set of numbers.

Measuring the Center

In a unimodal, symmetric distribution, these values will all be near the center.

In skewed distributions, they will be spread out.

Mean, Median, and Mode

If a distribution is symmetric, then the mean, median, and mode are all the same and are all at the center of the distribution.

Mean, Median, and Mode

However, if the distribution is skewed, then the mean, median, and mode are all different.

Mean, Median, and Mode

However, if the distribution is skewed, then the mean, median, and mode are all different.The mode is at the peak.

Mode

Mean, Median, and Mode

However, if the distribution is skewed, then the mean, median, and mode are all different.The mean is shifted in the direction of

skewing.

Mode Mean

Mean, Median, and Mode

ModeMedianMean

However, if the distribution is skewed, then the mean, median, and mode are all different.The median is (typically) between the mode

and the mean.

The Median vs. The Mean

If the data are strongly skewed, then the median is generally to give a more representative value.

If the data are not skewed, then the mean is usually preferred.

The Mean

Why is the average usually a good measure of the center?

If we have only two numbers, the average is half way between them.

What if we have more than two numbers? The mean balances the “deviations” on the

left with the “deviations” on the right.

The Mean

1 2 3 4 5 6 7 8 109

The Mean

1 2 3 4 5 6 7 8 109

Average

The Mean

1 2 3 4 5 6 7 8 109

Average

-2

-5

The Mean

1 2 3 4 5 6 7 8 109

Average

-2

-5

+1

+2

+4

The Median

1 2 3 4 5 6 7 8 109

The Median

1 2 3 4 5 6 7 8 109

Median

The Median

1 2 3 4 5 6 7 8 109

Median

-3

-6

The Median

1 2 3 4 5 6 7 8 109

Median

+1

+3

-3

-6

The Mean We use the letter x to denote a value from

the sample or population. The symbol means “add them all up.” So,

x means add up all the values in the population or sample (depending on the context).

Then the sample mean is n

x

The Mean

We denote the mean of a sample by the symbolx, pronounced “x bar”.

We denote the mean of a population by , pronounced “mu” (myoo).

Therefore,

N

xn

xx

TI-83 – The Mean Enter the data into a list, say L1. Press STAT > CALC > 1-Var Stats. Press ENTER. “1-Var-Stats” appears. Type L1 and press ENTER. A list of statistics appears. The first one is

the mean.

Case Study 8

2007 Small Arms Study (p. 47)Find the average number of guns per country

for India, China, Germany, France, and Pakistan.

Is the value representative of the group?

Then include the U.S. and compute the average for the six countries.

Is the value representative of the group?

The Median

1 2 3 4 5 6 7 8 109

The Median

1 2 3 4 5 6 7 8 109

Median

The Median Median – The middle value, or the average

of the middle two values, of a sample or population, when the values are arranged from smallest to largest.

The median, by definition, is at the 50th percentile. It separates the lower 50% of the sample from

the upper 50%.

The Median

When n is odd, the median is the middle number, which is in position (n + 1)/2.

When n is even, the median is the average of the middle two numbers, which are in positions n/2 and n/2 + 1.

Case Study 8

2007 Small Arms Study (p. 47)Find the median number of guns per country

for India, China, Germany, France, and Pakistan.

Is the value representative of the group?

Then include the U.S. and compute the median for the six countries.

Is the value representative of the group?

TI-83 – The Median

Follow the same procedure that was used to find the mean.

When the list of statistics appears, scroll down to the one labeled “Med.” It is the median.

TI-83 – The Median

Use the TI-83 to find the median number of guns.46, 40, 25, 19, 18.46, 40, 25, 19, 18, 270.

The Mode Mode – The value in the sample or

population that occurs most frequently. The mode is a good indicator of the

distribution’s central peak, if it has one.

Mode

The problem is that many distributions do not have a peak or they have several peaks.

In other words, the mode does not necessarily exist or there may be several modes.

Weighted Means

For the countries India, China, Germany, France, and Pakistan, the average number of guns per country is 29.6 million.

For Mexico and Brazil, the average is 15.4 million.

What is the average for all seven countries?

Weighted Means

The averages are India, et al: x1 = 29.6.

Mexico & Brazil: x2 = 15.4.

How could we combine the two averages to get the average for all seven countries?

Weighted Means

Compute the weighted average:

7

)4.15(2)6.29(5 x