chapter 3 numerical descriptive measures (3.1, 3.2, 3.3 ......

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3.1 Central TendencyMost variables show a distinct tendency to group around a central value. When people talk about an “average value” or the “middle value” or the “most frequent value,” they are talking informally about the mean, median, and mode—three measures of central tendency.

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! The central tendency is the extent to which all the data values group around a or central value.

! The variation is the amount of dispersion or scattering of values

! The shape is the pattern of the distribution of values from the lowest value to the highest value.

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Measures of Central TendencyThe Mean

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Y":4)&6'*)(3*D38&46" The location of the median when the values are in numerical

order (smallest to largest):

" If the number of values is odd, the median is the middle number

" If the number of values is even, the median is the average of the two middle numbers

Note that is not the value of the median, only the

position of the median in the ranked data

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" Value that occurs most often" Not affected by extreme values" Used for either numerical or categorical (nominal) data" There may may be no mode" There may be several modes

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Measures of Central Tendency:Review Example

House Prices:

$2,000,000$ 500,000$ 300,000$ 100,000$ 100,000

Sum $ 3,000,000

! Mean: ($3,000,000/5) = $600,000

! Median: middle value of ranked data

= $300,000! Mode: most frequent value

= $100,000

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Geometric mean! Used to measure the rate of change of a variable

over time

! Geometric mean rate of return! Measures the status of an investment over time

! Where Ri is the rate of return in time period i

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Example

An investment of $100,000 declined to $50,000 at the end of year one and rebounded to $100,000 at end of year two:

The overall two-year return is zero, since it started and ended at the same level.

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Use the 1-year returns to compute the arithmetic mean and the geometric mean:

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Arithmetic mean rate of return:

Geometric mean rate of return:

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Misleading result

More

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result

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Measures of Central Tendency:Summary

Central Tendency

Arithmetic Mean

Median Mode Geometric Mean

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3.2 Variation and ShapeMeasures of Variation

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The Range

! Simplest measure of variation! Difference between the largest and the smallest values:

Range = Xlargest – Xsmallest

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Example:

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Why The Range Can Be Misleading

! Does not account for how the data are distributed

! Sensitive to outliers

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/(((((0(((((1((((*2(((((**((((*+Range = 12 - 7 = 5

1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5

1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,120

Range = 5 - 1 = 4

Range = 120 - 1 = 119

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" Average (approximately) of squared deviations of values from the mean" >4A#?3 C4%&46:3O

The Sample Variance

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Xi = ith value of the variable X

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The Sample Standard Deviation

" Most commonly used measure of variation" Shows variation about the mean

" Is the square root of the variance" Has the same units as the original data

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Steps for Computing Standard Deviation1. Compute the difference between each value and the

mean.2. Square each difference.

3. Add the squared differences.4. Divide this total by n-1 to get the sample variance.5. Take the square root of the sample variance to get the

sample standard deviation.

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Calculation Example

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Comparing Standard Deviations

Mean = 15.5S = 3.338**((((*+((((*)((((*,((((*-((((*.((((*/((((*0((((*1((((+2(((+*

**((((*+((((*)((((*,((((*-((((*.((((*/((((*0((((*1((((+2(((+*

Data B

Data A

Mean = 15.5S = 0.926

**((((*+((((*)((((*,((((*-((((*.((((*/((((*0((((*1((((+2(((+*

Mean = 15.5S = 4.567

Data C

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Smaller standard deviation

Larger standard deviation

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Summary Characteristics

! The more the data are spread out, the greater the range, variance, and standard deviation.

! The more the data are concentrated, the smaller the range, variance, and standard deviation.

! If the values are all the same (no variation), all these measures will be zero.

! None of these measures are ever negative.

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Locating Extreme Outliers: Z-Score

where X represents the data valueX is the sample mean

S is the sample standard deviation

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Locating Extreme Outliers: Z-Score

! Suppose the mean math SAT score is 490, with a standard deviation of 100.

! Compute the Z-score for a test score of 620.

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A score of 620 is 1.3 standard deviations above the mean and would not be considered an outlier.

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Shape of a Distribution

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3.3 Exploring Numerical DataSections 3.1 and 3.2 discuss measures of central tendency, variation, and shape. You can also visualize the distribution of the values for a numerical variable by computing the quartiles and five-number summary and constructing aboxplot.Quartile

" Quartiles split the ranked data into 4 segments with an equal number of values per segment

" The first quartile, Q1, is the value for which 25% of the observations are smaller and 75% are larger

" Q2 is the same as the median (50% of the observations are smaller and 50% are larger)

" Only 25% of the observations are greater than the third quartile

i. i, i=

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Calculating The Interquartile Range

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The five numbers that help describe the center, spread and shape of data are:

! Xsmallest

! First Quartile (Q1)! Median (Q2)! Third Quartile (Q3)! Xlargest

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3.4 Numerical Descriptive Measures for a Population

! Descriptive statistics discussed previously described a sample, not the population.

! Summary measures describing a population, called parameters, are denoted with Greek letters.

! Important population parameters are the population mean, variance, and standard deviation.

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Measure Population Parameter

Sample Statistic

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Variance

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! Suppose that the variable Math SAT scores is bell-shaped with a mean of 500 and a standard deviation of 90. Then,

! 68% of all test takers scored between 410 and 590 (500 ± 90).

! 95% of all test takers scored between 320 and 680 (500 ± 180).

! 99.7% of all test takers scored between 230 and 770 (500 ± 270).

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