Measures of Cental Tendency

8/2/2019 Measures of Cental Tendency

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MEASURES OFCENTRAL TENDENCY

(AVERAGES)

Topic #6


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Measures

of Central Tendency

A measure of central tendencyis a univariate statistic

that indicates, in one manner or another,

the average ortypicalobserved value of a variable ina data set, or

put otherwise, the centerof the frequency distributionof the data.


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Central Tendency (cont.)

The central tendency of age among Berkeley faculty

member clearly increased from 1969-70 to 1988-89. On the other hand, the central tendency of Democratic

House vote by CD did not change much from 1948 to1970 (though other characteristics of its frequencydistribution certainly did change).


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How to Calculate Averages

We consider three measures of central tendency that areappropriate for three different levels of measurement(nominal, ordinal, and interva).

There are others, e.g., the geometricand harmonic

means, appropriate only for ratio variables.

For each measure of central tendency, we will considerhow it may be calculated from three different startingpoints: a list of observed values (i.e., raw data, e.g., one column in a

data spread sheet);

a frequency table (or bar graph);

a histogram and, in particular, a

continuous density curve (like income in PS #5C).


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The Mode

The mode (ormodal value) of a variable in a set of datais the value of the variable that is observed mostfrequentlyin that data (or, given a continuous frequencycurve, is at the point ofgreatest density).

Note: the mode is the value that is observed most

frequently, not the frequencyitself. (I see this error toofrequently on tests.)

The mode is defined foreverytype of variable [i.e.,

nominal, ordinal, interval, orratio]. However, the mode is used as a measure of central

tendency primarily for nominal variables only.


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The Mode (cont.) The mode is defined for ordinal and interval variables as well.

But since it takes account of only the nominal characteristics ofvariables (i.e., whether two cases have the same or different values), itis usually very unsatisfactory when applied to ordinal or higher levelvariables.

The mode may be ill-defined if we have either:

a small number of cases; or a precisely measured continuous variable and a finite number of

cases;

because in either event it is likely that no value will be observedmore than once in the data.

In any event, the mode is unstable in that

small changes in the data can result in large and erratic changes inthe modal value; and

changes in coding the variable (for example, RELIGION), orchanges in class intervals, can change the modal value.


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How To Calculate the Mode

Given a list of observed values (raw data):

Construct a frequency table (see next slide).


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How To Calculate the Mode (cont.)

Given a frequency table or bar graph (or having constructed one):

observe which value in the table (or graph) has the greatest(absolute orrelative) frequency;

the most frequent value (notthe frequencyitself) is the mode.

Notice that the modal number of problem sets turned in is 5, eventhough most students turned in fewer than 5,

So if we recoded the variable to create just two dichotomous categories:

(a) turned in all 5

(b) did not turn in all 5

the latter category becomes the modal category


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How To Calculate the Mode (cont.) Given a continuous frequency curve:

the mode is the value of the variable under thehighest pointof the frequency curve (the point withthe greatest density of observed values).


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The Median

The median (ormedian value) of a variable in a data setis the value in the middle of the observations, in the sense that no

more than halfof the cases have lowervalues and no more thanhalf of the cases have highervalues or,

more generally, such that no more than half of the cases havevalues that lie on either side of the median value.

Given a quite precisely measured continuous variableand a very large number of cases, we can in practice say

that half the cases have lower values and half have higher values

(e.g., LEVEL OF INCOME, SAT scores).

Equivalently, the median value is the value of the case at the50th percentile of the distribution.


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The Median (cont.)

G

iven a list of observed values (raw data): rank order the cases in terms of their observed values(e.g., from lowest to highest);

identify the value of the case right at the middle of thisrank-ordered list, and

the value of this case is the median value; or construct a frequency table and find where the

cumulative frequency crosses the 50% mark.


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The Median (cont.)


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The Median (cont.)If the number of cases is even, there is no observed value at the exact middle

of the list. Look at the pair of observations closest to the middle of the list. Ifthey have the same value, that value is the median. If they have differentvalues, every value in the intervalbounded by these two values meets thedefinition of a median; but conventionally the median in this event is definedas the midpointof the interval.


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TABLE 1 PERCENT OF POPULATION AGED 65 OR HIGHER IN THE 50 STATES

(UNIVARIATE DATA ARRAY)

Alabama 12.4 Montana 12.5

Alaska 3.6 Nebraska 13.8Arizona 12.7 Nevada 10.6

Arkansas 14.6 New Hampshire 11.5

California 10.6 New Jersey 13.0

Colorado 9.2 New Mexico 10.0

Connecticut 13.4 New York 13.0

Delaware 11.6 North Carolina 11.8

Florida 17.8 North Dakota 13.3

Georgia 10.0 Ohio 12.5Hawaii 10.1 Oklahoma 12.8

Idaho 11.5 Oregon 13.7

Illinois 12.1 Pennsylvania 14.8

Indiana 12.1 Rhode Island 14.7

Iowa 14.8 South Carolina 10.7

Kansas 13.6 South Dakota 14.0

Kentucky 12.3 Tennessee 12.4

Louisiana 10.8 Texas 9.7Maine 13.4 Utah 8.2

Maryland 10.7 Vermont 11.9

Massachusetts 13.7 Virginia 10.6

Michigan 11.5 Washington 11.8

Minnesota 12.6 West Virginia 13.9

Mississippi 12.1 Wisconsin 13.2

Missouri 13.8 Wyoming 8.9


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(1) Rank of state

(2) Name of state

(3) % over 65 instate (value ofvariable)

(4) percentile rankof state


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The Median (cont.)Is median income higher or lower than modal

income?


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The Median (cont.)Draw a vertical line such that half of the area under the

frequency curve lies on one side of the line and half onthe other side. The median value of the variable lies onthis line.


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Center of Gravity of Data The median (and the mode) may fail to be the center of

gravity or balance point of the data:Data (ranked): 5 6 7 8 14


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The Mean

The mean (ormean value) of a variable in a set of datais the result of adding up all the observed values of thevariable and dividing by the number of cases (i.e., theaverage as the term is most commonly used).

The mean is defined if and only if the variable is at least

intervalin nature [i.e., intervalorratio].

Suppose we have a variableXand a set of casesnumbered 1,2,...,n. Let the observed value of thevariable in each case be designated x1,x2, etc. Thus:


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How to Calculate the Mean

G

iven a list of observed values (raw data): Following the formula above, add up the observedvalues of the variable in all cases and divide by thenumber of cases.

Given a frequency table (or bar graph): Take each value and multiply it by its absolute

frequency, add up these products over all values, anddivide this sum by the number of cases; or(and this isusually easier)

take each value and multiply it by the decimal fraction(between zero and one) that represents its relativefrequency, and add up these products over all values.

Do not divide by the number of cases you have alreadydone this as a result of multiplying each value by a decimalfraction.


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The Mean (cont.) Is mean income lower or higher than median (or modal)

income?


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The Mean (cont.) The mean is the center of gravity of the distribution.

Determine (by eyeball approximation) the value of thevariable such that the density balances at that point;this value is the mean.


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Average Values For a quantitative variable, the range of observed values

of a variable in a set of data is the interval extendingfrom the minimum observed value to the maximumobserved value.

For everymeasure of central tendency, the averagevalue of the observations lies somewhere in this range,

often (but certainly not always) somewhere near themiddle of this range. The modal or median observed value can equal the minimum or

(like the modal problem sets turned in) the maximum value, but

the mean observation can do so only in the very special case in

which all cases have identical observed values, so there is nodispersion [next topic] in the data and min value = max value.

Once again, remember that the median is not (necessarily) themidpoint of the range.


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Average Values (cont.)

If a quantitative variable is discrete, so all of its valuesare whole numbers,

the modal or median observed value is always awhole number also (with the possible exception notedfor the median when the number of cases is even),

but the mean observation is almost never a whole

number.

The average family may have 2.374 children, i.e., the mean number of children per family may be 2.374,

even though no individual family can have that [or anyfractional] number of children.

Likewise, the mean number of problem sets turned inwas 3.85.


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Deviations From the Average

Unless all cases have the same observed value [i.e., nodispersion], some (in the case of mean, probably all)observed values will differ from the average value.

If the variable is interval (or ratio) in nature, we cancalculate the deviation from the average in each case, i.e., the distance or interval from the observed value to the

average value.

The deviation in each case is positive if the observed value is greater than the average value, negative if the observed value is less than the average value, or

zero if the observed value is equal to the average value.


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Deviations From the Average (cont.)

Since we have three different types of averages, we alsohave three different types of deviations from theaverage. The deviations from each type of averagehave different properties.

There are fewernon-zero deviations from the modal

value than from any other value of the variable. No more than halfof the deviations from the medianvalue arepositive and no more than halfare negative.

Unless several cases have the median value (andperhaps even then), the number of cases with

positive deviations equals the number with negativedeviations; that is,

the cases with positive and negative deviations balance outwith respect to theirnumber.


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Deviations From the Average (cont.)

The sum (or mean) of the absolute deviations (i.e.,ignoring whether the deviations are positive or negative)from the median is less than the sum (or mean) of theabsolute deviations from any other value of the variable.

The sum (or mean) of the (algebraic, i.e., takingaccount of whether deviations are positive or negative)deviations from the mean is zero; that is, the positive and negative deviations balance out with respect to

their total (positive and negative) magnitudes.

The sum (or mean) of the squared deviations from themean is less than the sum of the squared deviationsfrom any other value of the variable.


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Median vs. Mean Values (cont.)

If the distribution of the data issymmetric, the median and meanvalues are the same.

If the distribution is almostsymmetric, the median and themean are almost the same.

For example, test scores aretypically distributedapproximately symmetrically,so median and mean testscores are typically

approximately the same.Median Mean

MC 15.5 15.71

BB1 15.75 15.865

Score 32.75 31.577


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Median vs. Mean Values (cont.) If the distribution of the data is skewed, the mean is

pulled (relative to the median) in the direction of the longthin tail. For example, income is distributed in a highly skewed fashion,

with a long thin tail in the direction of higher income. Thus meanincome is typically considerably higher than median income.


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Question: which arrow (red or green) points tothe median value and which to the median?

On the course webpage, see Statistical Applets:

Mean and Median


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The median (unlike the mean) is resistant to outliers,where an outlier(in univariate data) is a case with anextreme (very high or very low) value. That is, addingsome outliers to the data (or removing them) may have abig impact on the mean value but usually has little impacton the median value.


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Median vs. Mean Values (cont.)If the observed values in some distribution of data changes, the

median value changes only ifthe value (or identity) of themedian case changes, whereas the mean value most likely isaffected by anychange in the data. For example, if the richget richer while everybody else stays about the same, meanincome increases while median income stays the same.


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If changes in the data do not change the sum of allobserved values, i.e., if the sum of all values remainsconstant, the mean remains constant, while the medianmay change.

For example, if Congress has decided that a tax cutwill total $100 billion but has not decided how thisfixed sum should be divided up among the nations100 million households,

the median benefit will depend on the specifics of

the legislation but the mean tax benefit per household will be

$1000 in any event.


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Median vs. Mean Values (cont.) If observed values arepolarizedwith approximately half

the cases having high values, half having low, and nonehaving medium values, the median is unstable; that is, the median value will be high or low depending on whether

slightly more than half the cases have high values or slightlymore than half the cases have low values, whereas

the mean value will barely change as a result of such small shiftsin the data.


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TEST 1: FREQUENCY DISTRIBUTION OF LETTER GRADES


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TEST 1: UNIVARIATE STATISTICS:

FALL 09 FALL 10


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TEST 1: BIVARIATE CHART AND SUMMARY STATISTIC

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Measures of Cental Tendency