Lecture 15 Sections 5.1 – 5.2 Wed, Sep 27, 2006 Measuring Center

Lecture 15Lecture 15

Sections 5.1 – 5.2Sections 5.1 – 5.2

Wed, Sep 27, 2006Wed, Sep 27, 2006

Measuring Measuring CenterCenter

Measuring the CenterMeasuring the Center Often, we would like to have Often, we would like to have oneone

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

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

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

Measuring the CenterMeasuring the Center

MeanMean – the simple average of a set – the simple average of a set of numbers.of numbers.

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

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

Measuring the CenterMeasuring the Center

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

In skewed distributions, they will be In skewed distributions, they will be spread out.spread out.

The MeanThe Mean

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

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

What if we have more than two What if we have more than two numbers?numbers?

The mean balances the “deviations” The mean balances the “deviations” on the left with the “deviations” on on the left with the “deviations” on the right.the right.

The MeanThe Mean

1 2 3 4 5 6 7 8 109

The MeanThe Mean

1 2 3 4 5 6 7 8 109

Average

The MeanThe Mean

1 2 3 4 5 6 7 8 109

Average

-2

-5

The MeanThe Mean

1 2 3 4 5 6 7 8 109

Average

-2

-5

+1

+2

+4

The MeanThe Mean

We use the letter We use the letter xx to denote a value to denote a value from the sample or population.from the sample or population.

The symbol The symbol means “add them all up.” means “add them all up.” So, So,

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

Then the sample mean isThen the sample mean isn

x

The MeanThe Mean

We denote the mean of a We denote the mean of a samplesample by the symbolby the symbolxx, pronounced “, pronounced “xx bar”.bar”.

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

Therefore,Therefore,

N

xn

xx

TI-83 – The MeanTI-83 – The Mean Enter the data into a list, say LEnter the data into a list, say L11.. Press STAT > CALC > 1-Var Stats.Press STAT > CALC > 1-Var Stats. Press ENTER. “1-Var-Stats” Press ENTER. “1-Var-Stats”

appears.appears. Type LType L11 and press ENTER. and press ENTER. A list of statistics appears. The first A list of statistics appears. The first

one is the mean.one is the mean.

ExamplesExamples

Use the TI-83 to find the mean of the Use the TI-83 to find the mean of the following data.following data.

2.942.9466

2.332.3355

3.413.4188

1.891.8900

2.732.7311

3.853.8555

1.341.3444

2.122.1266

2.882.8811

2.542.5422

2.502.5044

3.363.3677

1.951.9500

2.392.3922

2.442.4433

3.053.0533

The MedianThe Median

1 2 3 4 5 6 7 8 109


1 2 3 4 5 6 7 8 109

Median

The MedianThe Median MedianMedian – The middle value, or the – The middle value, or the

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

The median, by definition, is at the The median, by definition, is at the 5050thth percentile. percentile. It separates the lower 50% of the It separates the lower 50% of the

sample from the upper 50%.sample from the upper 50%.


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

When When nn is is eveneven, the median is the , the median is the average of the middle two numbers, average of the middle two numbers, which are in positions which are in positions nn/2 and /2 and nn/2 + /2 + 1.1.


Suppose we surveyed a two groups Suppose we surveyed a two groups of households and found the of households and found the following numbers of children:following numbers of children: Group 1: 3, 2, 5, 2, 6.Group 1: 3, 2, 5, 2, 6. Group Group 2: 2, 3, 0, 2, 1, 0, 3, 0, 1, 4.2: 2, 3, 0, 2, 1, 0, 3, 0, 1, 4.

Find the median number of children Find the median number of children in each group.in each group.

TI-83 – The MedianTI-83 – The Median

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

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

TI-83 – The MedianTI-83 – The Median

Use the TI-83 to find the medians of Use the TI-83 to find the medians of the samplesthe samples 3, 2, 5, 2, 63, 2, 5, 2, 6 2, 3, 0, 2, 1, 0, 3, 0, 1, 4 2, 3, 0, 2, 1, 0, 3, 0, 1, 4

The Median vs. The The Median vs. The MeanMean

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

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

The ModeThe Mode ModeMode – The value in the sample or – The value in the sample or

population that occurs most population that occurs most frequently.frequently.

The mode is a good indicator of the The mode is a good indicator of the distribution’s central peak, if it has distribution’s central peak, if it has one.one.

ModeMode

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

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

Weighted MeansWeighted Means

Find the average number of children Find the average number of children for each group. for each group. Group 1: 3, 2, 5, 2, 6.Group 1: 3, 2, 5, 2, 6. Group Group 2: 2, 3, 0, 2, 1, 0, 3, 0, 1, 4.2: 2, 3, 0, 2, 1, 0, 3, 0, 1, 4.

Weighted MeansWeighted Means

The averages areThe averages are Group 1: Group 1: xx11 = 3.6. = 3.6.

Group 1: Group 1: xx22 = 2.6. = 2.6.

How could we combine the two How could we combine the two averages to get the average of all averages to get the average of all the families together?the families together?

Mean, Median, and ModeMean, Median, and Mode

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


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


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

Mode


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

skewing.skewing.

Mode Mean


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

skewing.skewing. The median is (typically) between the The median is (typically) between the

mode and the mean.mode and the mean.

ModeMedianMean

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Lecture 15 Sections 5.1 – 5.2 Wed, Sep 27, 2006 Measuring Center