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Statistics with Economics and Business Applications
Statistics with Economics and Business Applications
Extreme Values
Extreme Values
The Range
The Range
The Variance
The Variance
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Statistics with Economics and Business Applications

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1Statistics with Economics and Business 21estimate the population standard deviation
Applications. Chapter 2 Describing Sets of s. Dividing by n –1 gives us a better
Data Descriptive Statistics – Numerical estimate of s.
Measures. 22Measures of Relative Standing. How
2Review. I. What’s in last lecture? many measurements lie below the
Descriptive Statistics – tables and measurement of interest? This is measured
graphs. Chapter 2. II. What's in this by the pth percentile. (100-p) %. p %.
lecture? Descriptive Statistics – 23Examples. 90% of all men (16 and
Numerical Measures. Read Chapter 2. older) earn more than $319 per week. ?
3Describing Data with Numerical Median. ? Lower Quartile (Q1). ? Upper
Measures. Graphical methods may not always Quartile (Q3). $319 is the 10th
be sufficient for describing data. percentile. BUREAU OF LABOR STATISTICS
Numerical measures can be created for both 2002.
populations and samples. A parameter is a 24Quartiles and the IQR. The lower
numerical descriptive measure calculated quartile (Q1) is the value of x which is
for a population. A statistic is a larger than 25% and less than 75% of the
numerical descriptive measure calculated ordered measurements. The upper quartile
for a sample. (Q3) is the value of x which is larger
4Measures of Center. A measure along than 75% and less than 25% of the ordered
the horizontal axis of the data measurements. The range of the “middle
distribution that locates the center of 50%” of the measurements is the
the distribution. interquartile range, IQR = Q3 – Q1.
5Some Notations. We can go a long way 25Calculating Sample Quartiles. The
with a little notation. Suppose we are lower and upper quartiles (Q1 and Q3), can
making a series of n observations. Then we be calculated as follows: The position of
write as the values we observe. Read as Q1 is. The position of Q3 is. once the
“x-one, x-two, etc” Example: Suppose we measurements have been ordered. If the
ask five people how many hours of they positions are not integers, find the
spend on the internet in a week and get quartiles by interpolation.
the following numbers: 2, 9, 11, 5, 6. 26Example. Q1is 3/4 of the way between
Then. the 4th and 5th ordered measurements, or
6Arithmetic Mean or Average. The mean Q1 = 65 + .75(65 - 65) = 65. The prices
of a set of measurements is the sum of the ($) of 18 brands of walking shoes: 60 65
measurements divided by the total number 65 65 68 68 70 70 70 70 70 70 74 75 75 90
of measurements. where n = number of 95. Position of Q1 = .25(18 + 1) = 4.75
measurements. Position of Q3 = .75(18 + 1) = 14.25.
7Example. Time spend on internet: 2, 9, 27Example. Q3 is 1/4 of the way between
11, 5, 6. If we were able to enumerate the the 14th and 15th ordered measurements, or
whole population, the population mean Q3 = 75 + .25(75 - 74) = 75.25. and IQR =
would be called m (the Greek letter “mu”). Q3 – Q1 = 75.25 - 65 = 10.25. The prices
8Median. The median of a set of ($) of 18 brands of walking shoes: 60 65
measurements is the middle measurement 65 65 68 68 70 70 70 70 70 70 74 75 75 90
when the measurements are ranked from 95. Position of Q1 = .25(18 + 1) = 4.75
smallest to largest. The position of the Position of Q3 = .75(18 + 1) = 14.25.
median is. once the measurements have been 28Using Measures of Center and Spread:
ordered. The Box Plot. Divides the data into 4 sets
9Example. The set: 2, 4, 9, 8, 6, 5, 3 containing an equal number of
n = 7 Sort: 2, 3, 4, 5, 6, 8, 9 Position: measurements. A quick summary of the data
.5(n + 1) = .5(7 + 1) = 4th. The set: 2, distribution. Use to form a box plot to
4, 9, 8, 6, 5 n = 6 Sort: 2, 4, 5, 6, 8, 9 describe the shape of the distribution and
Position: .5(n + 1) = .5(6 + 1) = 3.5th. to detect outliers. The Five-Number
10Mode. The mode is the measurement Summary: Min Q1 Median Q3 Max.
which occurs most frequently. The set: 2, 29Constructing a Box Plot. The
4, 9, 8, 8, 5, 3 The mode is 8, which definition of the box plot here is
occurs twice The set: 2, 2, 9, 8, 8, 5, 3 similar, but not exact the same as the one
There are two modes—8 and 2 (bimodal) The in the book. It is simpler. Calculate Q1,
set: 2, 4, 9, 8, 5, 3 There is no mode the median, Q3 and IQR. Draw a horizontal
(each value is unique). line to represent the scale of
11Example. The number of quarts of milk measurement. Draw a box using Q1, the
purchased by 25 households: 0 0 1 1 1 1 1 median, Q3.
2 2 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 5. Mean? 30Constructing a Box Plot. *. Isolate
Median? Mode? (Highest peak). outliers by calculating Lower fence:
12Extreme Values. The mean is more Q1-1.5 IQR Upper fence: Q3+1.5 IQR
easily affected by extremely large or Measurements beyond the upper or lower
small values than the median. The median fence is are outliers and are marked (*).
is often used as a measure of center when 31Constructing a Box Plot. Draw
the distribution is skewed. “whiskers” connecting the largest and
13Extreme Values. Symmetric: Mean = smallest measurements that are NOT
Median. Skewed right: Mean > Median. outliers to the box.
Skewed left: Mean < Median. 32Example. Amount of sodium in 8 brands
14Measures of Variability. A measure of cheese: 260 290 300 320 330 340 340
along the horizontal axis of the data 520.
distribution that describes the spread of 33Example. IQR = 340-292.5 = 47.5 Lower
the distribution from the center. fence = 292.5-1.5(47.5) = 221.25 Upper
15The Range. The range, R, of a set of n fence = 340 + 1.5(47.5) = 411.25. Outlier:
measurements is the difference between the x = 520.
largest and smallest measurements. 34Interpreting Box Plots. Median line in
Example: A botanist records the number of center of box and whiskers of equal
petals on 5 flowers: 5, 12, 6, 8, 14 The length—symmetric distribution Median line
range is. R = 14 – 5 = 9. Quick and easy, left of center and long right
but only uses 2 of the 5 measurements. whisker—skewed right Median line right of
16The Variance. The variance is measure center and long left whisker—skewed left.
of variability that uses all the 35Key Concepts. I. Measures of Center 1.
measurements. It measures the average Arithmetic mean (mean) or average a.
deviation of the measurements about their Population mean: m b. Sample mean of size
mean. Flower petals: 5, 12, 6, 8, 14. n: 2. Median: position of the median =
17The Variance. The variance of a .5(n +1) 3. Mode 4. The median may be
population of N measurements is the preferred to the mean if the data are
average of the squared deviations of the highly skewed. II. Measures of Variability
measurements about their mean m. The 1. Range: R = largest - smallest.
variance of a sample of n measurements is 36Key Concepts. 2. Variance a.
the sum of the squared deviations of the Population of N measurements: b. Sample of
measurements about their mean, divided by n measurements: 3. Standard deviation.
(n – 1). 37Key Concepts. IV. Measures of Relative
18The Standard Deviation. In calculating Standing 1. pth percentile; p% of the
the variance, we squared all of the measurements are smaller, and (100 - p)%
deviations, and in doing so changed the are larger. 2. Lower quartile, Q 1;
scale of the measurements. To return this position of Q 1 = .25(n +1) 3. Upper
measure of variability to the original quartile, Q 3 ; position of Q 3 = .75(n
units of measure, we calculate the +1) 4. Interquartile range: IQR = Q 3 - Q
standard deviation, the positive square 1 V. Box Plots 1. Box plots are used for
root of the variance. detecting outliers and shapes of
19Two Ways to Calculate the Sample distributions. 2. Q 1 and Q 3 form the
Variance. Use the Definition Formula: 5. ends of the box. The median line is in the
-4. 16. 12. 3. 9. 6. -3. 9. 8. -1. 1. 14. interior of the box.
5. 25. Sum. 45. 0. 60. 38Key Concepts. 3. Upper and lower
20Two Ways to Calculate the Sample fences are used to find outliers. a. Lower
Variance. Use the Calculational Formula: fence: Q 1 - 1.5(IQR) b. Outer fences: Q 3
5. 25. 12. 144. 6. 36. 8. 64. 14. 196. + 1.5(IQR) 4. Whiskers are connected to
Sum. 45. 465. the smallest and largest measurements that
21Some Notes. The value of s is ALWAYS are not outliers. 5. Skewed distributions
positive. The larger the value of s2 or s, usually have a long whisker in the
the larger the variability of the data direction of the skewness, and the median
set. Why divide by n –1? The sample line is drawn away from the direction of
standard deviation s is often used to the skewness.
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