Chapter 3 – Chapter 3 – Data DescriptionData Description

Section 3.1 – Section 3.1 – Measures of Central TendencyMeasures of Central Tendency

Measures of Central TendencyMeasures of Central Tendency“’“’Average’ when you stop to think about it is a funny Average’ when you stop to think about it is a funny concept. Although it describes all of us it describes concept. Although it describes all of us it describes none of us…While none of us wants to be the average none of us…While none of us wants to be the average American, we all want to know about him or her.”American, we all want to know about him or her.”

The average American man is 5’9” tall.The average American man is 5’9” tall.The average American woman is 5’3.6”.The average American woman is 5’3.6”.The average American is sick in bed 7 days a year The average American is sick in bed 7 days a year missing five days of work.missing five days of work.On the average day, 24 million people receive animal On the average day, 24 million people receive animal bitesbitesBy his or her 70By his or her 70thth birthday, the average American will birthday, the average American will have eaten 14 steers, 1050 chickens, 2.5 lambs and have eaten 14 steers, 1050 chickens, 2.5 lambs and 25.2 hogs.25.2 hogs.

Measures of Central TendencyMeasures of Central TendencyAverage does not mean exactly what you think Average does not mean exactly what you think it means…it means…

An average could indicate several different An average could indicate several different thingsthings MeanMean MedianMedian ModeMode MidrangeMidrangeAlso, an ‘”average” could be describe a Also, an ‘”average” could be describe a samplesample or a or a populationpopulation

Measures of Central TendencyMeasures of Central TendencyStatistic – Statistic – a characteristic or measure obtained by using data a characteristic or measure obtained by using data

values from a SAMPLE. values from a SAMPLE. We generally use ROMAN letters to represent We generally use ROMAN letters to represent

statistics (A, B, C, D …)statistics (A, B, C, D …)

Parameter – Parameter – a characteristic or measure obtained by using data a characteristic or measure obtained by using data

values from a POPULATION.values from a POPULATION. We generally use GREEK letters to represent We generally use GREEK letters to represent

parameters (parameters (σσ, , ββ, , αα, , µµ …) …)

Measures of Central TendencyMeasures of Central TendencyMean – Mean – the sum of the values, divided by the total number of the sum of the values, divided by the total number of

values.values. this is usually what you are talking about when you this is usually what you are talking about when you

say averagesay average

x-bar is used to represent the x-bar is used to represent the samplesample mean. mean.µ is used to represent the µ is used to represent the populationpopulation mean. mean.

1 2 1... n nX X X X XXn n

Measures of Central TendencyMeasures of Central TendencyThe Median (MD)The Median (MD)

arrange observations from smallest to arrange observations from smallest to largest.largest.median is either the middle number or the median is either the middle number or the mean of the middle two numbers.mean of the middle two numbers.

Measures of Central TendencyMeasures of Central TendencyThe ModeThe Mode the data value that occurs most frequentlythe data value that occurs most frequently UnimodalUnimodal- data set has only one value with the - data set has only one value with the

greatest frequency.greatest frequency. BimodalBimodal- data set has 2 values with the greatest - data set has 2 values with the greatest

frequency.frequency. MultimodalMultimodal- data set has more than 2 values with - data set has more than 2 values with

the greatest frequency.the greatest frequency. No modeNo mode- no data value occurs more than once- no data value occurs more than once

Measure of Central TendencyMeasure of Central Tendency

Modal Class- class with the largest Modal Class- class with the largest frequencyfrequency

Pg. 112 example 3-12Pg. 112 example 3-12

Measures of Central TendencyMeasures of Central TendencyThe MidrangeThe Midrange the mid point of a data setthe mid point of a data set MR= (min + max) / 2MR= (min + max) / 2

Measures of Central TendencyMeasures of Central TendencyRounding Rules – Rounding Rules – In general, round to one place after the last In general, round to one place after the last

place given in the data.place given in the data.

ex. 3.45, 5.21, 6.89, 4.22ex. 3.45, 5.21, 6.89, 4.22 round to three decimal places.round to three decimal places.

Measures of Central TendencyMeasures of Central TendencyThe Median and Mode are The Median and Mode are resistantresistant, , meaning unusually large or small values meaning unusually large or small values do not affect it.do not affect it.The Mean and Midrange are not. The one The Mean and Midrange are not. The one huge house in the neighborhood allows huge house in the neighborhood allows the mean home value to skyrocket.the mean home value to skyrocket.

Measures of Central TendencyMeasures of Central TendencyGROUPED DATAGROUPED DATA

When data is grouped in a distribution or When data is grouped in a distribution or in a graph things are slightly different.in a graph things are slightly different.First, make a distribution table with these First, make a distribution table with these column headingscolumn headings ClassClass FrequencyFrequency MidpointMidpointThen add one more column Then add one more column frequency times midpointfrequency times midpoint

Measures of Central TendencyMeasures of Central TendencyGROUPED DATAGROUPED DATA

Look at the procedure table on pg. 108Look at the procedure table on pg. 108

Finding the mean for grouped dataFinding the mean for grouped data

Measures of Central TendencyMeasures of Central TendencyGROUPED DATAGROUPED DATA

Turn to page 107 and look at example 3-3.Turn to page 107 and look at example 3-3. Remember: Midpoint = Upper limit + lower Remember: Midpoint = Upper limit + lower

limit / 2limit / 2

You will add your f●xYou will add your f●xmm altogether and then altogether and then divide by the total frequency (n). divide by the total frequency (n).

Measures of Central TendencyMeasures of Central TendencyGROUPED DATAGROUPED DATA

Weighted MeanWeighted Mean found by multiplying each value by its found by multiplying each value by its

corresponding “weight” and dividing the sum of the corresponding “weight” and dividing the sum of the products by the sum of the weights.products by the sum of the weights.

For exampleFor example Grade Point Averages in CollegeGrade Point Averages in College A = 4 points, B = 3 points, C = 2 points, D = 1 point.A = 4 points, B = 3 points, C = 2 points, D = 1 point. Each class has a different weight…Each class has a different weight…

Example 3-17 pg. 115Example 3-17 pg. 115Class Credits (w) Grade (x) wXEnglish Composition 1 3 A (4 points)

Introduction to Psychology 3 C (2 points)

Biology 1 4 B (3 points)

Physical Education 2 D(1 point)

X = ΣwXΣw

=

Distribution Shapes Distribution Shapes (pics pg. 59)(pics pg. 59)Bell-Shaped- a single peak and tapers off at Bell-Shaped- a single peak and tapers off at either endeither endUniform- flat or rectangularUniform- flat or rectangularJ-Shaped- few data values on left side, J-Shaped- few data values on left side, increases from left to rightincreases from left to rightReverse J-shaped- few data values on right Reverse J-shaped- few data values on right side, decreases from left to rightside, decreases from left to rightBimodal- has 2 peaks of the same heightBimodal- has 2 peaks of the same heightU-shaped- shaped like a UU-shaped- shaped like a U

Distribution Shapes (pg. 117 for Distribution Shapes (pg. 117 for pics)pics)

Positively Skewed or Right-SkewedPositively Skewed or Right-Skewed Majority of data values fall on the left of the mean and Majority of data values fall on the left of the mean and

cluster at the lower endcluster at the lower end Tail is to the rightTail is to the right

Symmetric DistributionSymmetric Distribution Data values evenly distributed on both sides of the Data values evenly distributed on both sides of the

meanmeanNegatively Skewed or Left-SkewedNegatively Skewed or Left-Skewed Data values fall on right of the mean and cluster at the Data values fall on right of the mean and cluster at the

upper end.upper end. Tail is to the leftTail is to the left

Practice!Practice!

Pg. 118Pg. 118 2, 3, 7, 10, 12, 132, 3, 7, 10, 12, 13