1. Numerical Descriptive Statistics
Business Statistics Topic 3
Numerical Descriptive Statistics

2. Business Statistics:Topic 3
Learning Objectives
By the end of this topic you will be able to:
Describe and apply the measures of central location: mean, mode and median.
Determine the significance of the skewness of a distribution
Describe and apply non-central locations: Quartiles
Describe and apply the measures of Variation: range, interquartile range and standard deviation
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3. Business Statistics:Topic 3
Thinking
What is descriptive statistics?
How can we get the various features of a set of data
How can we summarize the data?
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4. How would you judge which salesperson should receive the bonus?
The management of a company is looking at performances of two of their salespersons to determine to give them a bonus.
Sales Person A
Sales person B
Month
Number of sales
Jan
1100
800
Feb
1400
1800
Mar
1200
600
Apr
1300
1900
May
2000
Jun
1500
500
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5. Business Statistics:Topic 3
To calculate the average?
Salesperson A salesperson B 1267
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6. Measures of Central Tendencies
Generally a data set shows a distinct tendency to group about a certain central point or location. These central locations are:
Mean
Mode
Median
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7. Business Statistics:Topic 3
Mean
Definition:
The arithmetic average and the most common measure of central tendency.
e.g. average rate of return on different investments
e.g. average waiting time for service
Note:
All values are included in computing the mean.
A set of data has a unique mean
The mean is affected by unusually large or small data points (outliers / extreme values).
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8. Business Statistics:Topic 3
Sample Mean
Raw Data
Where ‘x’ is data and ‘n’ is the total number of data
Example
The annual incomes of middle-management employees at Westinghouse are:
$42,900, $49,100, $38,300, and $56,800
The average income is given by
(42, , , ,800)/4= 46,775
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9. Business Statistics:Topic 3
Effect of Outliers
Mean of
is 6
(sorted data)
Mean of data when an extreme value is included
is 8
(sorted data)
Note : mean is shifted to the right because of 29 (outlier-high extreme value)
What would happen to the mean if you added a low
extreme value?
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10. Business Statistics:Topic 3
Properties
The mean is unique for a given set of data.
The disadvantage with the mean is that it is affected by the presence of extreme or unusual values we can outliers.
85, 42, 89, 88, 89, 79, 95, 90, 93, 82
So if there are outliers or extreme values then the valuess of the arithmetic mean will not give you an accurate idea of the location of most of the values. Thus the mean loses its representativeness of the data.
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11. Business Statistics:Topic 3
Mode
Definition:The most frequent data, or data corresponding to the highest frequency.
Note:
Mode is not affected by extreme values
There may not be a mode
There may be several modes
Used for either numerical or categorical data
Example:
Raw Data
Mode = 5
(5 occurs the most frequently of all the values in the raw data)
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12. Business Statistics:Topic 3
properties
The mode can be used with either numerical or categorical data.
The mode for a given set of data is not unique. Sometimes you may not have any mode or more than one mode.
Mode is not affected by outliers.
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13. Business Statistics:Topic 3
Median
The Median is the value that splits a ranked set fo data into two equal parts.
Middle observation after the data is sorted (ascending or descending order).
1. If ‘n’ is odd, the median is the value of the (n+1)/2th ordered data.
2. If ‘n’ is even, the median is the mean of the n/2th and (n+1)/2th ordered data
Note: Median is not affected by extremely large or small values and is therefore a valuable measure of central tendency when such values occur.
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14. Calculating the Median
Example:
Sort data in ascending order:
Here n(the amount of numbers)=7 (note: an odd no.).
Calculate the middle, or 4th position: (7+1)/2=4th position.
The 4th observation is 5 therefore the median is 5.
Sort data first:
Here n=8 (Note: an even number)
There are two middle positions: n/2 and the next one. These are the 4th and 5th positions. The 4th observation is 5 and the 5th is 7
Calculate the average of 5 and 7. (5+7)/2=6
Therefore the median is 6
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15. Business Statistics:Topic 3
Effect of Outlier
Median of
is 5
(sorted data)
is 5
(sorted data)
Note : median is not affected by 29 (outlier)
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16. Applying the Central Tendencies
Mean:
When the data (polygon) is asymmetrical, or skewed, the mean loses its representativeness.
Median:
With skewed distributions, the median is the preferred measure of central tendency. The skew would need to be very strong, however, before the mean is rejected.
Mode:
When there is no mode or more than two values of mode, it is not a useful representation of data. However, the mode is very suitable for qualitative data.
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17. Non- Central Locations: Quartiles
Calculating quartiles:
Divide data into four equal parts
The use of quartile calculations:
Quartiles are often used with scores for aptitude tests
They are also used in commerce and industry when a large number of observations is involved.
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18. Business Statistics:Topic 3
Quartiles
First Quartile ( ): There are 25% of the
observations below and 75% above.
Second Quartile ( ): There are 50% of the
observations below and 50% above.
Third Quartile ( ): There are 75% of the
observations below and 25% above. Q 2
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19. Procedure for finding Quartiles
Suppose n/4 is not a whole number.
Let ‘m’ be the next whole number larger than n/4.
Then the lower quartile is the mth observation of the sorted data counting from the lower end.
The upper quartile is mth observation of the sorted data counting from the upper end.
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20. Procedure for finding Quartiles (continued)
Suppose n/4 is a whole number.
Let ‘m’ be =n/4. Then the lower quartile is average of the mth and (m+1)th observation of the sorted data counting from the lower end.
The upper quartile is similarly defined, counting from the upper end.
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21. Business Statistics:Topic 3
Quartiles: example
Consider the data :
Sort data first
Here n=7, 7/4=1.75, which is not a whole number. The next whole number is 2. Therefore the 2 nd observation in the sorted data from the lower end is which is 2.
2nd observation in the sorted data from the upper end is , which is 8.
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22. Business Statistics:Topic 3
2. Measures of Variation
In addition to measures of central tendencies, it is often desirable to consider measures of dispersion.
Suppose you are looking at performances of sales persons who have equal sales average, it would be difficult to assess who has performed better. Only the spread of the data could tell us who is consistent in generating sales.
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23. Business Statistics:Topic 3
Start Thinking
Start thinking
How would you judge which
Salesperson should receive the
bonus?
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24. 2 Measures of variability 变异度 离散度
The measures of dispersion we will be discussing are:
Range 全距
Inter-quartile Range 四分位数间距
Standard Deviation 标准差
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25. Business Statistics:Topic 3
Observing and Thinking
Observe these two
data sets:
Small variability
The average value provides
a good representation of the
observations in the data set.
This data set is now
changing to...
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26. Measures of variability
Observe two hypothetical
data sets:
Small variability
The average value provides
a good representation of the
observations in the data set.
Larger variability
The same average value does not
provide as good representation of the
observations in the data set as before.
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27. 2.1 Range 全距
The range of a set of observations is the difference between the largest and smallest observations.
Its major advantage is the ease with which it can be computed.
Its major shortcoming is its failure to provide information on the dispersion of the observations between the two end points.
But, how do all the observations spread out? ? ? ?
The range cannot assist in answering this question
Range
Smallest
observation
Largest
observation
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28. Business Statistics:Topic 3
Range 全距
The range measures the total spread in the set data.
For example: Range could be used to measure the following:
the temperature fluctuation in a given day movement of share prices.
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29. Effect of Outliers 离群值的影响
The Range is affected by the presence of outliers.
It does not take into account how the data is distributed between the highest and the smallest values
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30. Interquartile Range 四分位数间距
The Interquartile Range is defined as the difference between the upper and lower quartiles.
IQR = —
This measures the spread in the middle 50% of the data
It is not affected by outlying observations
It does not consider how the data is distributed
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31. Business Statistics:Topic 3
2.3 Standard Deviation
Definition of Standard Deviation:
A measure of the variation of data from the mean.
The most commonly used measure of variation
Represented by the symbol ‘s’
Shows how the data is distributed around the mean
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32. Calculation of SD from a sample
where
X sample data
- sample mean
n sample size
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33. Business Statistics:Topic 3
Example
Here = 81
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34. Comparing Standard Deviations
Data A
Mean = 13.5
s = 3.338
Data B
Mean = 13.5
s = .9258
Data C
Mean = 13.5
s = 4.57
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35. % of data within one sd of mean
% of data which falls in the interval
Approximately 67% of the data lies within one standard deviation of the mean.
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36. Business Statistics:Topic 3
Sample Variance
Sample variance is the square of the sample
standard deviation
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37. Understanding Variation in Data
The more spread out, or dispersed the data are, the larger will be the range, interquartile range and the standard deviation
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38. Relative location—Z score
The mean and standard deviation can be used together to learn about the relative locations, z of observations in a data.
Where is the sample mean and ‘s’ is the sample standard deviation.
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39. Business Statistics:Topic 3
Shape of distribution
What factors influence the distribution of a set of data?
The shape of a distribution is determined by the relative positions of the central tendencies.
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40. Business Statistics:Topic 3
Shape of Distribution
Symmetric Distribution
Mode = Median = Mean
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41. Right Skewed Distribution
A small portion of relatively large extreme values pull the polygon to the right.
Note: Skewness is a measure of asymmetry.
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42. Left Skewed Distribution
A small portion of extreme low values pull the polygon to the left.
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43. The Five-Number Summary
Mean
Mode
Median
Lower Quartile
Upper Quartile
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44. Business Statistics:Topic 3
Box-and-whisker Plot
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45. Symmetrical Distribution
The median at equal distance from either quartile
The whiskers have the same length
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46. Symmetrical Distribution
Consider the following data:
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47. Business Statistics:Topic 3
Skewed to The Left
The median is closer to the third quartile
The left whisker is much longer than the right whisker
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48. Skewed to the Left: example
Consider the following data:
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49. Business Statistics:Topic 3
Skewed to The Right
The median is closer to the first quartile
The right whisker is much longer than the left whisker
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50. Skewed to the Right: example
Consider the following data:
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51. Assessing the strength of skewness
A set of data is said to be significantly skewed
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52. Business Statistics:Topic 3
Population Measures
Mean
Standard Deviation
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53. Ethical Considerations
Results needs to be given in a:
fair,
objective, and
neutral manner
Learn to differentiate between poor presentation
and unethical presentation.
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54. Business Statistics:Topic 3
Summary
In this topic you have:
Looked at calculating & applying the measures of central tendencies : mean, mode and median
Reviewed the concept of central tendency and the shape of distributions
Looked at calculating & applying the measures of dispersion : range, interquartile range and standard deviation
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55. Business Statistics:Topic 3
Main points
preview what we have learned
Raise question about the variation-lead in
Summer measure of variation: standard deviation: definition, property, calculation; application and meaning
Visual presentation: bell-shaped; box-whisker
Using excel: standard deviation, box and whisker
Final Summary
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56. Business Statistics:Topic 3全距
标准差
标准分
分布情况
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