L2 Descriptive Statistics

IELM151/ Stuart X. Zhu 1

Lecture 2 Descriptive Statistics

• Source of Data• Types of Data• Graphical methods for describing a set of

data• Numerical methods for describing a set of

data• Summary• Readings: Chap. 1

Source of Data

• Data distributed by an organization or an individual – HK Census and Statistics Department

• A survey• A designed experiment• An observational study


http://www.censtatd.gov.hk/home/index.jsp


Types of Data

• Quantitative– Numerical, computable, describes quantity– E.g., height, weight, salary, cost, time, distance

• Qualitative– Nonnumerical, categorical, describes an

attribute– E.g., blood type, gender, grading, professional

rank

Types of DataData

Categorical Numerical

Discrete ContinuousExamples:

Marital StatusPolitical PartyEye Color(Defined categories)

Examples:

Number of ChildrenDefects per hour(Counted items)

Examples:

WeightVoltage

(Measured characteristics)


Graphical Approaches

• (Relative) Frequency Histogram

• Stem and Leaf Plot



Company % Company % Company %

1 9.4 11 7.5 21 11.12 8.4 12 10.2 22 8.53 12.5 13 9.9 23 9.44 6.7 14 8.2 24 9.75 10 15 8.8 25 12.36 7.8 16 11.7 26 10.67 10.2 17 7.9 27 8.98 9.5 18 10.3 28 8.19 6.5 19 7.5 29 6.910 11.4 20 11 30 10

PR Example: Here are the data concerning the percentages of revenue (PR) spent on R&D for 30 HK companies.


(Relative) Frequency HistogramPrinciples for constructing histograms:

• Determine the number of classes– Sturges’ Formula: k = 1 + 3.3 × log10 (n)

• Determine class width– Approximate to (Maximum – Minimum ) / k

• Locate the class boundaries– Start from the lowest class boundary which is smaller than the

minimum and locate the others one by one. Note that a measurement cannot fall on a boundary. (See “L02_Descriptive_s1.xls”)

Usage of Relative Frequency Histogram– Proportion of PR > 10.05% in the 30 HK companies (Sample)– Estimate “the fraction of PR > 10.05% for all HK companies”

(Population)

Stem and Leaf Plot

• A stem-and-leaf plot organizes data into groups (called stems) so that the values within each group (the leaves) branch out to the right on each row.



Stem-and-leaf Plot of PR Example

Stem Leaf

6 579

7 5589

8 124589

9 44579

10 0 02236

11 0147

12 35

Frequency

3

4

6

5

6

4

2

Split each sample into two parts consisting of a stem and a leaf


Stem and Leaf Plot

• Rotate the plot counterclockwise 90o

(Relative) Frequency Histogram versus Stem and Leaf Plot

012345678

6.55 7.55 8.55 9.55 10.55 11.55 12.55Fr

eque

ncy

Percentage of Revenue

Frequency Histogram


Double-Stem-and-leaf Plot of PR ExampleStem Leaf

6 5797 55898. 1248* 5899. 449* 57910. 0022310* 611. 01411* 712. 312* 5

Frequency

343323513111


Numerical Approaches

• Parameters– Numerical descriptive measures computed from

POPULATION measurements• Statistics

– Numerical measures computed from SAMPLE measures

• Facts– Parameters are constant though they may be

unknown– Statistics change from sample to sample (random)


Measure of Location (Central Tendency, (Center of the Distribution)

• Mean– Sample mean– Population mean– Sample: 5, 15, 7, 34, 450

• Median– The middle value of xi’s (even / odd)– Can be obtained from the stem

and leaf plot

∑

∑

=

=

=+++

=

=+++

=

N

ii

N

n

ii

n

xNn

xxx

xnn

xxxx

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⎪⎩

⎪⎨⎧

+=

+

+

even isn If),(21

odd isn If,~

12/2/

2/)1(

nn

n

xx

xx

• Mode– The value that appears most frequently in all

the xi’s– Sample: 3, 7, 24, 5, 9, 11, 13, 15, 66, 66– Modal Class in (Relative) Frequency

Histogram



Skewness of Data• Three types

– Symmetric:mean = median

– Skewed to the right:mean > median

– Skewed to the left:mean < median

• PR Example– mean = 9.36 <

median = 9.45A little bit skewed to the left

• Remark– More skewed, the

differences among the measures of central tendency become greater

– When skewed or contain extreme values, median& mode are better descriptions.

– Advantage of mean over median & mode

• More amenable to mathematical & theoretical treatment

• More stable if n is large


Measure of Variability (Dispersion of the Distribution)

• Range = Maximum – Minimum– Sample 1: 5, 15, 7, 34, 30– Sample 2: 5, 19, 18, 20, 34

• Measure of deviation from the mean– Mean Absolute Deviation (MAD)– Variance

• Population variance• Sample variance

∑=

−=n

ii xx

nMAD

1||1

∑=

−−

=n

ii xx

ns

1

22 )(1

1

∑=

−=N

iix

N 1

22 )(1 μσ


An Exercise• Exercise 1:

(a) Sample 1: 12, 6, 15, 3, 8, 7, 10, 10, 9, 9. Compute mean, median, range, MAD, and sample variance.(b) Sample 2: 12, 6, 15, 3, 4, 7, 10, 15, 9, 7. Do the same calculation. (See “L02-04_Descriptive_s2-1.xls”)

• An important and useful ‘short-cut’ formula

)1()(

11

2

11

2

1

22

−

⎟⎠

⎞⎜⎝

⎛−

=−−

=∑∑

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= nn

xxnxx

ns

n

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iin

ii


Another Exercise

• Exercise 2:Sample: 17, 5, 4, 10, 2, 11Verify that mean = 8.1667, median = 7.5, s=5.5648, n=6, Now, the and the maximum of three additional data are respectively 75, 991, and 20. What are the sample mean, sample median, and the sample standard deviation of the combined set of 9 data?


Measure of Relative Variation• Coefficient of variation CV (for positive

measurements only)

• Sample A: 3, 10, 7, 4, 6, mean =6, s = 2.7386, CV = 45.64%

• Sample B: 30, 100, 70, 40, 60, mean = 60, s= 27.386, CV = 45.64%

• Sample C: 13, 20, 17, 14, 16, mean = ?, s = ?, CV =?

%100×=xsCV


Measure of Relative Standing

• Percentile– Population: The pth percentile is the value of x such

that p% of the measurements are less than that value of x and (100 – p) % greater.

– Sample: – Lower Quartile (QL) 25%, Upper Quartile (QU) 75%

• Sample 1: 12, 6, 15, 3, 8, 7, 10, 10, 9, 9Sorted: 3, 6, 7, 8, 9, 10, 10, 12, 15QL = (10 + 1)/4 th obs.

= (6 + 7)/2 = 6.5QU = (10 + 1)×3/4 th obs.

= (10 + 12)/2 = 11


Concepts based on Percentile

• Interquartile range = QU –QL

• Trimmed mean– The mean of ‘trimmed’ sample by eliminating data

below pth percentile and above (1 – p) th percentile• Sample 1: 12, 6, 15, 3, 8, 7, 10, 10, 9, 9

– Interquartile range = QU – QL = 10.5 – 6.75 = 3.75– The 10% trimmed mean is the mean of the following

sample 12, 6, 8, 7, 10, 10, 9, 9, which is 8.875


Z-Score (Standard Score)• The sample z-score corresponding to a particular

observation x is

• Criterion – 2<|z-score|<3 is quite likely– |z-score|>3 is very unlikely

• If |z-score|>2, the observation is a possible OUTLIER.• Sample: 5, 15, 27, 14, 20, 35, 27, 450; |z-score| = 2.47

for the last data; check the source of data before further analysis

sxxscorez −

=−


Summary

• Graphical methods are good in presenting data, not easy for comparison, and difficult to use of statistical inference.

• Numerical methods mainly focus on the CENTRAL VALUE and the SPREAD of data.

• Different measures have their own advantages and disadvantages. Be careful and smart in using them.


Cathay Pacific’s Experiment

• The marketing group wanted to increase the number of business class seats sold on its off-peak flights. Key factors were identified as advertising level and pricing strategy. There exist two levels of advertising campaigns and three pricing strategies in geographically.

• Question– Which level and strategy are the best?


334 Reform

• Effect on HKUST undergraduate education– The development of students’ ability– Employment opportunity


Documents

L2 Descriptive Statistics