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1 Class Session #2 Numerically Summarizing Data Measures of Central Tendency Measures of Dispersion Measures of Central Tendency and Dispersion from Grouped Data Measures of Position

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2 Recall the Definitions Parameter – a descriptive measure of a population (p = parameter = population, usually in Greek letters) Statistic – a descriptive measure of a sample (s = statistic = sample, usually in Roman letters)

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3 Common “descriptions” ? Average ? – “typical” as described in the news reports Give some of today’s examples Data distributions’ “characteristics” –Shape – look at a picture (histogram) –Center – mean, mode, median –Spread – range, variance, std. dev.

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4 Central Tendency Definitions Arithmetic mean – the sum of all the values of the variable in the data set, divided by the number of observations Population arithmetic mean - computed using all the individuals in the population (“mew” = μ) (≠ micro µ) Sample arithmetic mean – computed using the sample data (“x-bar”) Note: is a statistic, μ is a parameter

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5 More Central Tendency Defs Median – the value that lies in the middle of the data, when arranged in ascending order (think of the median strip of highway in the middle of the road) Mode – the most frequent observation of the variable in the data set (think “a la mode” in fashion /on top)

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6 Measures of Dispersion Definitions Range (R) – the difference between the largest data value (maximum) & the smallest data value (minimum) Deviation about the mean – how “spread out” the data is. ? for both population and sample variance, the sum of all deviations about the mean equals what ? ? the square of a non-zero number is ?

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7 More Measures of Dispersion Definitions Population Variance – sum of squared deviations about the population mean, divided by the number of observations in the population N (sigma squared) ? i.e. population variance is the mean of the ______ _________ ____ __ _________ ___ ? Answer: Population variance is the mean of the squared deviations about the population mean

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8 More Measures of Dispersion Definitions Sample Variance – sum of the squared deviations about the sample mean, divided by the number of observations minus one (s squared) Degrees of freedom is the “n-1”

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9 More Measures of Dispersion Definitions Population Standard Deviation – the square root of the population variance (sigma, written as “σ”) Sample Standard Deviation – the square root of the sample variance (s, written as “s”) BTW, later we discover “s” itself is a random variable

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10 Empirical Rule for Symmetric Data If the distribution is bell shaped: 68% of data within 1 std deviations 95% of data within 2 std deviations 99.7% of data within 3 standard deviations of the mean Rule holds for both samples & populations

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11 Supposing Grouped Data Approximate mean of a variable from a frequency distribution Use the midpoint of each class Use the frequency of each class Use the number of classes Population Mean Sample Mean

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12 Supposing Grouped Data Weighted Mean Good to use when certain data values have higher importance (or weight) [Sum of each value of variable times its weight] / [sum of weights] Examples of Grade Point Average (GPA) and mixed nuts pricing

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13 Supposing Grouped Data Population Variance sum of [(midpoint – mean) 2 times frequency] / [sum of frequencies] Sample Variance as before except “-1” in denominator (the degrees of freedom thing again)

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14 Supposing Grouped Data Population Standard Deviation take square root of population variance Sample Standard Deviation take square root of sample variance

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15 Measures of Position Definition z-Score – the distance that a data value is from the mean in terms of standard deviations. Equals (data value minus mean) divided by standard deviation] Population z-score Sample z-score

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16 Measures of Position Definitions z-score equals [(data value minus mean) divided by standard deviation] Is a "unitless" measure Can be “normalized” to get Mean of zero Standard Deviation of one

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17 M easures of Position Definition s z-score purpose is to provide a way to "compare apples and oranges" by converting variables with different centers and/or spreads to variables with the same center (0) and spread (1).

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18 Measures of Position Definition Percentiles – k th percentile is a set of data divides the lower k% from the upper (1-k)% Divide into 100 parts, so 99 percentiles exist “P sub k” Use to give relative standing of the data

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19 Measures of Position Definition Quartiles – divides the data into four equal parts Four parts, so three percentiles exist “Q sub one, two, or three” Q 2 is the median of the data Q 1 is the median of the lower half Q 3 is the median of the upper half

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20 Numerical summary of data Five number summaries Interquartile range (Q 3 – Q 1) is resistant to extreme values Compute five number summary Min value | Q 1 | M | Q 3 | max value

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21 Building a Box Plot – part 1 1. Calculate interquartile range (IQR) 2. Compute lower & upper fence Lower fence = Q 1 – 1.5 (IQR) Upper fence = Q (IQR) 3. Draw scale then mark Q 1 and Q 3 4. Box in Q 1 to Q 3 then mark M

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22 Building a Box Plot – part 2 5. Temporarily mark fences with brackets 6. Draw line from Q 1 to smallest value inside the lower fence and a line from Q 3 to largest value inside the upper fence 7. Put * for all values outside of the fences 8. Erase brackets

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23 Distribution based on Boxplot Symmetric median near center of box horizontal lines about same length Skewed Right / Positive Skew median towards left of box right line much longer than left line Skewed Left / Negative Skew median towards right of box left line much longer than right line

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24 Which measure best to report? Symmetric distribution Mean Standard Deviation Skewed distribution Median Interquartile Range

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25 Self Quiz When can the mean and the median be about equal? In the 2000 census conducted by the U.S. Census Bureau, two average household incomes were reported: $41,349 and $55,263. One of these averages is the mean and the other is the median. Which is which and why?

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26 Self Quiz The U.S. Department of Housing and Urban Development (HUD) uses the median to report the average price of a home in the United States. Why do they do that?

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27 Self Quiz A histogram of a set of data indicates that the distribution of the data is skewed right. Which measure of central tendency will be larger, the mean or the median? Why?

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28 Self Quiz If a data set contains 10,000 values arranged in increasing order, where is the median located? Matching: (parameter; statistic) _____ is a descriptive measure of a population _____ is a descriptive measure of a sample.

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29 Self Quiz A data set will always have exactly one mode. (true or false) If the number of observations, n, is odd ; then the median, M, is the value calculated by the formula M=(n+1)/2

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30 Self Quiz Find the Sample Mean: 20, 13, 4, 8, 10 Find the Sample Mean: 83, 65, 91, 87, 84 Find the Population Mean: 3, 6, 10, 12, 14

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31 Self Quiz The median for the given list of six data values is , 12, 21,, 41, 50 What is the missing value?

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32 Self Quiz The following data represent the monthly cell phone bill for the cell phone for six randomly selected months. $35.34$42.09 $39.43 $38.93 $43.39$49.26 Compute the mean, median, and mode cell phone bill.

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33 Self Quiz Heather and Bill go to the store to purchase nuts, but can not decide among peanuts, cashews, or almonds. They agree to create a mix. They bought 2.5 pounds of peanuts for $1.30 per pound, 4 pounds of cashews for $4.50 per pound, and 2 pounds of almonds for $3.75 per pound. Determine the price per pound of the mix.

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