chap 3-1 statistics for business and economics topic 3 describing data: numerical figures won´t...
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Chap 3-1Statistics for Business and Economics
Topic 3
Describing Data: Numerical
Figures won´t lie, but liars will figure
Why descriptive statistics is important to managers?
Managers also need to become acquainted with numerical descriptive measures that provide very brief and easy-to understand summaries of a data collection
There are two broad categories into which these measures fail: measures of central tendency and measures of variability
Statistics for Business and Economics Chap 3-3
After completing this topic, you should be able to compute and interpret the:
Measures of central tendency, variation, and shape (Arithmetic) Mean, median, mode, geometric mean Quartiles Five number summary and box-and-whisker plots Range, interquartile range, variance and standard
deviation, coefficient of variation Symmetric and skewed distributions Flat and peaked distribution
Topic Goals
Statistics for Business and Economics Chap 3-4
Describing Data Numerically
Arithmetic Mean
Median
Mode
Describing Data Numerically
Variance
Standard Deviation
Coefficient of Variation
Range
Interquartile Range
Central Tendency Variation
Geometric Mean
Statistics for Business and Economics Chap 3-5
Measures of Central Tendency
Central Tendency
Mean Median Mode
n
xx
n
1ii
Overview
Midpoint of ranked values
Most frequently observed value
Arithmetic average
Statistics for Business and Economics Chap 3-6
Arithmetic Mean
The arithmetic mean (mean) is the most common measure of central tendency
For a population of N values:
For a sample of size n:
Sample size
n n
i 1 i 1x (simple) or x (weighted)i i ix x n
n n
Observed
values
N N
ii 1 i 1μ (simple) or μ = (weighted)
i ix x n
N N
Population size
Population values
Arithmetic mean
The arithmetic mean of a collection of numerical values is the sum of these values divided by the number of values. The symbol for the population mean is the Greek letter μ (mu), and the symbol for a sample mean is X (X-bar)
A population parameter is any measurable characteristic of a population.
A sample statistic is any measurable characteristic of a sample.
Characteristics of the arithmetic mean
l. Every data set measured on an interval or ratio level has a mean.
2. The mean has valuable mathematicaI properties that make it convenient to use in further computations.
3. The mean is sensitive to extreme values. 4. The sum of the deviations of the numbers in a
data set from the mean is zero 5. The sum of the squared deviations of the
numbers in a data set from the mean is a minimum value.
Statistics for Business and Economics Chap 3-9
Geometric Mean
The geometric mean is the most common measure of central tendency for rates (growth rates, interest rates, etc.)
For N values:
NNN21
N
1iigeo xxxxμ
Statistics for Business and Economics Chap 3-10
Arithmetic Mean
The most common measure of central tendency Mean = sum of values divided by the number of values Affected by extreme values (outliers) !!!
(continued)
0 1 2 3 4 5 6 7 8 9 10
Mean = 3
0 1 2 3 4 5 6 7 8 9 10
Mean = 4
35
15
5
54321
4
5
20
5
104321
Statistics for Business and Economics Chap 3-11
Median
The numerical value in the middle when data set is arranged in order (50% above, 50% below)
Not affected by extreme values
0 1 2 3 4 5 6 7 8 9 10
Median = 3
0 1 2 3 4 5 6 7 8 9 10
Median = 3
Statistics for Business and Economics Chap 3-12
Finding the Median
The location of the median:
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
dataorderedtheinposition2
1npositionMedian
2
1n
Statistics for Business and Economics Chap 3-13
Mode
A measure of central tendency Value that occurs most often Not affected by extreme values Used for either numerical or categorical data There may be no mode There may be several modes
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Mode = 9
0 1 2 3 4 5 6
No Mode
Statistics for Business and Economics Chap 3-14
Review Example:Summary Statistics
Mean: (€3,000,000/5)
= €600,000
Median: middle value of ranked data = €300,000
Mode: most frequent value = €100,000
House Prices:
€2,000,000 500,000 300,000 100,000 100,000
Sum 3,000,000
Statistics for Business and Economics Chap 3-15
Mean is generally used, unless extreme values (outliers) exist
Then median is often used, since the median is not sensitive to extreme values. Example: Median home prices may be
reported for a region – less sensitive to outliers
Which measure of location is the “best”?
Statistics for Business and Economics Chap 3-16
Same center, different variation
Measures of Variability
Variation
Variance Standard Deviation
Coefficient of Variation
Range Interquartile Range
Measures of variation give information on the spread or variability of the data values.
Statistics for Business and Economics Chap 3-17
Range
Simplest measure of variation Difference between the largest and the smallest
observations:
Range = Xlargest – Xsmallest
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Range = 14 - 1 = 13
Example:
Statistics for Business and Economics Chap 3-18
Ignores the way in which data are distributed
Sensitive to outliers
7 8 9 10 11 12
Range = 12 - 7 = 5
7 8 9 10 11 12
Range = 12 - 7 = 5
Disadvantages of the Range
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
Statistics for Business and Economics Chap 3-19
Quartiles
Quartiles split the ranked data into 4 segments with an equal number of values per segment
25% 25% 25% 25%
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% are smaller, 50% are larger)
Only 25% of the observations are greater than the third quartile
Q1 Q2 Q3
Statistics for Business and Economics Chap 3-20
Quartile Formulas
Find a quartile by determining the value in the appropriate position in the ranked data, where
First quartile position: Q1 = 0.25(n+1)
Second quartile position: Q2 = 0.50(n+1) (the median position)
Third quartile position: Q3 = 0.75(n+1)
where n is the number of observed values
Statistics for Business and Economics Chap 3-21
(n = 9)
Q1 = is in the 0.25(9+1) = 2.5 position of the ranked data
so use the value half way between the 2nd and 3rd values,
so Q1 = 12.5
Quartiles
Sample Ranked Data: 11 12 13 16 16 17 18 21 22
Example: Find the first quartile
Statistics for Business and Economics Chap 3-22
Interquartile Range
Can eliminate some outlier problems by using the interquartile range
Eliminate high- and low-valued observations and calculate the range of the middle 50% of the data
Interquartile range = 3rd quartile – 1st quartile
IQR = Q3 – Q1
Statistics for Business and Economics Chap 3-23
Interquartile RangeFive number summary –Box plot
Median(Q2)
XmaximumX
minimum Q1 Q3
Example:
25% 25% 25% 25%
12 30 45 57 70
Interquartile range = 57 – 30 = 27
Statistics for Business and Economics Chap 3-24
Average of squared deviations of values from the mean Population variance:
Population Variance
N N2 2
i i2 2i 1 i 1
(x μ) (x μ)σ (simple) or σ (weighted)
N N
in
Where = population mean
N = population size
xi = ith value of the variable x
ni = absolute frequency
μ
Statistics for Business and Economics Chap 3-25
Average (approximately) of squared deviations of values from the mean Sample variance:
Sample Variance
n n2 2
i i2 2i 1 i 1
(x x) (x x)s (simple) or s (weighted)
n-1 n-1
in
Where = arithmetic mean
n = sample size
xi = ith value of the variable x
ni = absolute frequency
x
Statistics for Business and Economics Chap 3-26
Population Standard Deviation
The square root of population variance Most commonly used measure of variation Shows variation about the mean Has the same units as the original data
Population standard deviation:
N N2 2
i ii 1 i 1
(x μ) (x μ)σ (simple) or σ (weighted)
N N
in
Statistics for Business and Economics Chap 3-27
Sample Standard Deviation
The square root of the sample variance Most commonly used measure of variation Shows variation about the mean Has the same units as the original data
Sample standard deviation:n n
2 2i i
i 1 i 1
(x x) (x x) (simple) or (weighted)
n-1 n-1
ins s
Statistics for Business and Economics Chap 3-28
Calculation Example:Sample Standard Deviation
Sample Data (xi) : 10 12 14 15 17 18 18 24
n = 8 Mean = x = 16
4.24267
126
18
16)(2416)(1416)(1216)(10
1n
)x(24)x(14)x(12)X(10s
2222
2222
A measure of the “average” scatter around the mean
Statistics for Business and Economics Chap 3-29
Measuring variation
Small standard deviation
Large standard deviation
Statistics for Business and Economics Chap 3-30
Comparing Standard Deviations
Mean = 15.5 s = 3.338 11 12 13 14 15 16 17 18 19 20 21
11 12 13 14 15 16 17 18 19 20 21
Data B
Data A
Mean = 15.5 s = 0.926
11 12 13 14 15 16 17 18 19 20 21
Mean = 15.5 s = 4.570
Data C
Statistics for Business and Economics Chap 3-31
Advantages of Variance and Standard Deviation
Each value in the data set is used in the calculation
Values far from the mean are given extra weight (because deviations from the mean are squared)
Statistics for Business and Economics Chap 3-32
Coefficient of Variation
Measures relative variation Always in percentage (%) Shows variation relative to mean Can be used to compare two or more sets of
data measured in different units
scv 100%
x
Statistics for Business and Economics Chap 3-33
Comparing Coefficient of Variation
Stock A: Average price last year = $50 Standard deviation = $5
Stock B: Average price last year = $100 Standard deviation = $5
Both stocks have the same standard deviation, but stock B is less variable relative to its price
A
s $5100% 100% 10%
x $50cv
B
s $5100% 100% 5%
x $100cv
Statistics for Business and Economics Chap 3-34
Skewness
Shows asymmetry and refers to the shape of a distribution
Can take on positive, negative or zero values
n n3 3
i ii 1 i 1
1 13 3
(x x) (x x) (simple) or (weighted)
in
s n s n
Distribution Shape
The shape of the distribution is said to be symmetric if the observations are balanced, or evenly distributed, about the center; coefficient of skewness equal zero
0123456789
10
1 2 3 4 5 6 7 8 9
Fre
qu
en
cy
Symmetric Distribution
Chap 3-35Statistics for Business and Economics
When the distribution is unimodal, the mean, median, and mode are all equal to one another and are located at the center of the distribution
Distribution Shape
The shape of the distribution is said to be skewed if the observations are not symmetrically distributed around the center.
(continued)
Positively Skewed Distribution
0
2
4
6
8
10
12
1 2 3 4 5 6 7 8 9
Fre
qu
ency
Negatively Skewed Distribution
0
2
4
6
8
10
12
1 2 3 4 5 6 7 8 9
Fre
qu
ency
A positively skewed distribution (skewed to the right) has a tail that extends to the right in the direction of positive values.
A negatively skewed distribution (skewed to the left) has a tail that extends to the left in the direction of negative values.
Chap 3-36Statistics for Business and Economics
Statistics for Business and Economics Chap 3-37
Shape of a Distribution
Describes how data are distributed Measures of shape
Symmetric or skewed
Mean = Median = Mode Mean < Median < Mode Mean <Median < Mode
Right-SkewedLeft-Skewed Symmetric
Kurtosis
Refers to the shape of a distribution Can take positive (peaked distribution), negative
(flat distribution) or zero values (a symmetrical, bell-shaped, normal distribution)
Statistics for Business and Economics Chap 3-38
n n4 4
i ii 1 i 1
2 24 4
(x x) (x x) - 3 (simple) or -3 (weighted)
in
s n s n
Kurtosis
Statistics for Business and Economics Chap 3-39
(continued)
Statistics for Business and Economics Chap 3-40
If the data distribution is bell-shaped, then the interval:
contains about 68% of the values in the population or the sample
The Empirical Rule
1σμ
μ
68%
1σμ
Statistics for Business and Economics Chap 3-41
contains about 95% of the values in the population or the sample
contains about 99.7% of the values in the population or the sample
The Empirical Rule
2σμ
3σμ
3σμ
99.7%95%
2σμ
Statistics for Business and Economics Chap 3-42
Using Microsoft Excel
Simple Descriptive Statistics can be obtained from Microsoft® Excel
Use menu choice:
Data Tab / Data analysis / Descriptive statistics
Enter details in dialog box
Statistics for Business and Economics Chap 3-43
Enter dialog box details
Check box for summary statistics
Click OK
Using Microsoft Excel(continued)
Statistics for Business and Economics Chap 3-44
Excel output
Microsoft Excel
Simple descriptive statistics output, using the house price
data:
House Prices:
$2,000,000 500,000 300,000 100,000 100,000
Next topic
Theory of probability
Thank you!
Have a nice day!