1. MBA 510 Lecture 2
Spring 2013
Dr. Tonya Balan
4/20/2019

2. 4/20/2019

3. Data Summary Descriptive Measures
A descriptive measure is a single number computed from the sample data that provides information about the data. The four types of descriptive measures are:
Measures of central tendency – Where is the middle of my data?
Measures of variation – How spread out are my data?
Measures of position – Where does my value fall within the distribution?
Measures of shape – Are my data symmetric?
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4. Measures of Central Tendency
Mean - average
Median – center value
Midrange – half-way point
Mode – most frequently occurring value
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5. Data Summary Descriptive Measures
A descriptive measure is a single number computed from the sample data that provides information about the data. The four types of descriptive measures are:
Measures of central tendency – Where is the middle of my data?
Measures of variation – How spread out are my data?
Measures of position – Where does my value fall within the distribution?
Measures of shape – Are my data symmetric?
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6. Measures of Variation
A single statistic is generally not sufficient to completely describe the distribution of the data
Some data are more homogeneous than others
Variation is the tendency of the data values to scatter about the sample mean
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7. Measures of Variation
Range – distance between the smallest and largest values
Variance – describes the variation of the sample values around the sample mean
Standard Deviation – square root of the variance
Coefficient of Variation (CV) – measure of the standard deviation relative to the mean
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8. Range
The range is the numerical distance between the smallest and largest value in the data set
As such, the range is highly influenced by outliers in the data
Range = H - L
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9. Range Data 37 40 28 H = 42, L=28 Range = H – L = 42 – 28 = 14 35 36 4230 31 33 39
H = 42, L=28
Range = H – L = 42 – 28 = 14
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10. Range Data 37 40 28 H = 58, L=28 Range = H – L = 58 – 28 = 30 35 36 4231 33 39 58
H = 58, L=28
Range = H – L = 58 – 28 = 30
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11. Variance
The variance measures the spread (or variation) of the sample values about the sample mean.
To calculate the variance
Compute the distance from each sample value to the sample mean
Square the distances
Sum up the squared distances
Divide by (n-1) where n is the sample size
𝑠 2 = (𝑥− 𝑥 ) 2 𝑛−1
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12. Example
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13. Standard Deviation
The standard deviation also measures the spread of the sample values about the sample mean. However, the unit of measure for the standard deviation is the same as the unit of measure for the data, so it is easier to interpret than the variance.
The standard deviation is simply the square root of the variance
s= (𝑥− 𝑥 ) 2 𝑛−1
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14. Additional Comments
Because the units of measure are the same for the mean and the standard deviation, we can use the information to ask questions such as:
“How many of the sample values are within two standard deviations of the mean?”
The magnitude of the variance (or standard deviation) is relative the magnitude of the data.
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15. Coefficient of Variation
To compare the variation across different samples, a relative measure of variation is needed
The coefficient of variation (CV) provides a measure of the standard deviation as a percentage of the mean
𝐶𝑉= 𝑠 𝑥
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16. Levi’s Example PT1 PT2 PT3 PT4 PT5 Mean 4.52 8.83 4.83 7.49 10.37
Median
1.95
6.15
4.7
7.1
11.3
Std. Deviation
10.03
15.35
4.4
3.66
9.56
C.V.
2.22
1.74
0.91
0.49
0.92
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17. Levi’s Example PT1 PT2 PT3 PT4 PT5 Mean 4.52 8.83 4.83 7.49 10.37
Median
1.95
6.15
4.7
7.1
11.3
Std. Deviation
10.03
15.35
4.4
3.66
9.56
C.V.
2.22
1.74
0.91
0.49
0.92
4/20/2019

18. Data Summary Descriptive Measures
A descriptive measure is a single number computed from the sample data that provides information about the data. The four types of descriptive measures are:
Measures of central tendency – Where is the middle of my data?
Measures of variation – How spread out are my data?
Measures of position – Where does my value fall within the distribution?
Measures of shape – Are my data symmetric?
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19. Measures of Position
Measures of Position are indicators of how a particular value fits in with all the other data values
Percentiles (and quartiles)
Z-score
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20. Percentiles
The pth percentile is the value such that exactly p% of the data are less than that value
“She scored in the 95th percentile of all students taking the test. In other words, 95% of the students taking the test had scores lower than hers. Only 5% of the students scored better than she did.”
The 50th percentile is also known as the median.
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21. Quartiles
The 25th, 50th, and 75th percentiles are also referred to as the quartiles of the distribution
Q1 = 1st quartile = 25th percentile
Q2 = 2nd quartile = 50th percentile
Q3 = 3rd quartile = 75th percentile
The quartiles will be important for constructing box and whisker plots.
Interquartile range (IQR)
IQR = Q3 – Q1
The IQR is the range that contains the middle 50% of the data
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22. Z-Score
The Z-Score expresses the position of a data value, x, in terms of the number of standard deviations above or below the mean.
𝑧= 𝑥 − 𝑥 𝑠
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23. Data Summary Descriptive Measures
A descriptive measure is a single number computed from the sample data that provides information about the data. The four types of descriptive measures are:
Measures of central tendency – Where is the middle of my data?
Measures of variation – How spread out are my data?
Measures of position – Where does my value fall within the distribution?
Measures of shape – Are my data symmetric?
4/20/2019

24. Skewness
The Skewness coefficient (Sk)measures the degree of skewness in your data
Value of Sk ranges from -3 to 3.
Sk = 0 for perfectly symmetric distributions
Sk<0 the data are skewed left
Sk>0 the data are skewed right
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25. Kurtosis Kurtosis measures the “peakedness” of your distribution
The value of the kurtosis is large if there is a high frequency of observations near the mean and in the tails of the distribution
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26. Chebyshev’s Inequality
For Any Data Set:
At least 75% of the data values are between 𝑥 −2𝑠 and 𝑥 +2𝑠 . At least 75% of the data values have a z-score between -2 and 2.
At least 89% of the data values are between 𝑥 −3𝑠 and 𝑥 +3𝑠 . At least 89% of the data values have a z-score between -3 and 3.
In general, at least (1− 1 𝑘 2 )𝑥100% of your data values lie between 𝑥 −𝑘𝑠 and 𝑥 +𝑘𝑠 (have z-scores between –k and k) for any k>1 .
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27. Empirical Rule
For data that have a normal distribution (bell-shaped distribution with skewness near 0):
Approximately 68% of the data values are between 𝑥 −𝑠 and 𝑥 +𝑠 .
Approximately 95% of the data values are between 𝑥 −2𝑠 and 𝑥 +2𝑠 .
Approximately 99.7% of the data values are between 𝑥 −3𝑠 and 𝑥 +3𝑠 .
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28. Box Plots
A Box Plot is a graphical representation of a set of sample data that illustrates the lowest data value (L), the first quartile (Q1), the median (Q2), the third quartile (Q3), the interquartile range, and the highest value (H).
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29. Box Plots
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30. Box Plots
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31. Box Plots
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32. Describing Bivariate Datax
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33. Describing Bivariate Data
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34. Describing Bivariate Data
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35. Correlation
The Pearson coefficient of correlation (r) measures the strength of the linear relationship between two variables and has the following important properties:
r ranges from -1 to 1
The larger |r| is, the stronger the linear relationship
r near zero indicates that there is no linear relationship
r=1 or r=-1indicates a perfect linear relationship
The sign of r tells you whether the relationship is positive or negative (e.g., as x increases, does y increase or decrease)
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36. Describing Bivariate Datax
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37. Understanding Stock Rates
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38. Correlation does not imply causation!
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