Describing Data: Numerical Measures

Population Parameter and Sample Statistics
Describing Data:
Numerical Measures
z
統計上用來描述資料特性的摘要性數值，稱之
為統計表徵數 (Statistical Characteristic ) 或統計
測量數 (Statistical Measurement)。
z
依描述母體與樣本的不同，分別稱之為：
Chapter 3
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McGraw-Hill/Irwin
©The McGraw-Hill Companies, Inc. 2008
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樣本統計量 (Sample Statistic)
母體參數 (Population Parameter)
z
母體參數是描述母體資料特性的統計測量數，一般簡
稱為參數或母數。參數是統計想要獲得的數值，是統
計的核心。
z
母體參數是個固定的常數。
z
但因常常無法直接求得，因此必須使用統計技巧，由
樣本資料估計。
z
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母體參數 (Population Parameter)
------ 用來描述母體特質的測量數。
樣本統計量 (Sample Statistic)
------ 用來描述樣本特質的測量數。
z
樣本統計量是描述樣本資料特性的統計測量數，一般
簡稱為統計量。
z
樣本統計量可由樣本觀察值計算而得。
z
樣本統計量常用來推論母體參數。
z
例如：樣本平均數、樣本變異數、樣本標準差。
例如：母體平均數、變異數、標準差、、、、
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常用統計表徵數
z
Center of Location (中心位置): 測量母體或樣本中心位置
的指標。
z
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Dispersion (分散程度)： 測量母體或樣本元素離散程度的
指標。
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Range（全距）
Mean Deviation（平均離差）
Variance（變異數）
Standard Deviation（標準差）
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Mean
Median
Mode
Geometric Mean
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Population Mean
Characteristics of the Mean
For ungrouped data, the population mean is the
sum of all the population values divided by the
total number of population values:
The arithmetic mean is the most widely used measure
of location. It requires the interval scale. Its major
characteristics are:
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Center of Location (中心位置):測量母體
或樣本中心位置的指標。
Mean（算數平均數）
Median（中位數）
Mode（眾數）
Geometric Mean（幾何平均數）
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z
Center of Location
All values are used.
It is unique.
The sum of the deviations from the mean is 0.
It is calculated by summing the values and dividing
by the number of values.
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EXAMPLE – Population Mean
Sample Mean
z
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EXAMPLE – Sample Mean
Properties of the Arithmetic Mean
z
z
z
z
z
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For ungrouped data, the sample mean is the
sum of all the sample values divided by the
number of sample values:
Every set of interval-level and ratio-level data has a mean.
All the 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 values.
The arithmetic mean is the only measure of central tendency
where the sum of the deviations of each value from the mean
is zero.
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Weighted Mean
z
EXAMPLE – Weighted Mean
The Carter Construction Company pays its hourly employees
$16.50, $19.00, or $25.00 per hour. There are 26 hourly employees,
14 of which are paid at the $16.50 rate, 10 at the $19.00 rate, and 2
at the $25.00 rate. What is the mean hourly rate paid the 26
employees?
The weighted mean of a set of numbers
X1, X2, ..., Xn, with corresponding weights
w1, w2, ...,wn, is computed from the
following formula:
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The Median
z
The Median is the midpoint of the values
after they have been ordered from the
smallest to the largest.
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Properties of the Median
z
There is a unique median for each data set.
z
It is not affected by extremely large or small values and
is therefore a valuable measure of central tendency
when such values occur.
There are as many values above the median as
below it in the data array.
z
It can be computed for ratio-level, interval-level, and
ordinal-level data.
For an even set of values, the median will be the
arithmetic average of the two middle numbers.
z
It can be computed for an open-ended frequency
distribution if the median does not lie in an open-ended
class.
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EXAMPLES - Median
The ages for a sample of
five college students are:
21, 25, 19, 20, 22
Arranging the data in
ascending order gives:
19, 20, 21, 22, 25.
Thus the median is 21.
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The Mode
z
The heights of four
basketball players, in
inches, are:
76, 73, 80, 75
The mode is the value of the observation
that appears most frequently.
Arranging the data in
ascending order gives:
73, 75, 76, 80.
Thus the median is 75.5
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Example - Mode
Mean vs. Median
z
z
當分配是對稱時 (Symmetric)，Mean 與 Median 是相
同的。
當分配是傾斜的 (Skewed)，Mean 與 Median 有較大
差別。
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可以把 Mean 當成重心，當分配向右傾斜時，重心會向右，
而 Median 對每個元素所給的重量（權重）是相同的，所以
不受極端值影響。
X={1,2,3,4,5} Y={1,2,3,4,90}
Mean(X) = 3 Median(X)=3
Mean(Y) = 20 Median(Y) =3
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The Relative Positions of the Mean,
Median and the Mode
The Geometric Mean
z
z
z
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Useful in finding the average change of percentages,
ratios, indexes, or growth rates over time.
It has a wide application in business and economics
because we are often interested in finding the
percentage changes in sales, salaries, or economic
figures, such as the GDP, which compound or build on
each other.
The geometric mean will always be less than or equal
to the arithmetic mean.
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The Geometric Mean
z
z
EXAMPLE – Geometric Mean
Suppose you receive a 5 percent increase in
salary this year and a 15 percent increase next
year. The average annual percent increase is
9.886, not 10.0. We begin by calculating the
geometric mean.
The geometric mean of a set of n positive numbers is
defined as the nth root of the product of n values.
The formula for the geometric mean is written:
GM = ( 1.05 )( 1.15 ) = 1.09886
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EXAMPLE – Geometric Mean (2)
EXAMPLE – Geometric Mean (3)
The return on investment earned by Atkins construction
Company for four successive years was: 30 percent, 20
percent, -40 percent, and 200 percent. What is the geometric
mean rate of return on investment?
The prices of a stock for three successive days were:
100, 60 , 90.
What is the geometric mean rate of return of this investment?
GM = 4 ( 1.3 )( 1.2 )( 0.6 )( 3.0 ) = 4 2.808 = 1.294
GM = 2 ( 0.6 )( 1.5 ) − 1 = 2 0.9 − 1 = −5.13%
If you compute the arithmetic mean [30+20-40+200]/4 = 52.5,
you would overstate the mean rate of return.
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If you compute the arithmetic mean [-40%+50%]/2 = 5%,
you would overstate the mean rate of return.
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Why Study Dispersion?
Dispersion or Variability （分散程度）
z
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Dispersion：測量母體或樣本元素離散程度的
指標。
– Range
– Mean Deviation
– Variance
– Standard Deviation
z
A measure of location, such as the mean or the
median, only describes the center of the data. It is
valuable from that standpoint, but it does not tell
us anything about the spread of the data.
z
For example, if your nature guide told you that
the river ahead averaged 3 feet in depth, would
you want to wade across on foot without
additional information? Probably not. You would
want to know something about the variation in the
depth.
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Why Study Dispersion?
Why Study Dispersion?
• A measure of dispersion can be used to evaluate the reliability
of two or more measures of location.
• Mean = 4.9, but the spread of the data is from 6m to 16.8y.
• The average is not representative because of the large spread.
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Mean Deviation
Range
Mean Deviation:
The arithmetic mean of the absolute values of the
deviations from the arithmetic mean.
Advantage:
1. All data are used in the computation.
2. Easy to understand : average of deviations
form the mean.
Drawback:
--- Absolute values are difficult to work with.
Example:
The number of cappuccinos sold at the Starbucks location in the
Orange Country Airport between 4 and 7 p.m. for a sample of 5 days
last year were 20, 40, 50, 60, and 80. Determine the range for the
number of cappuccinos sold.
Range = Largest – Smallest value
= 80 – 20 = 60
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EXAMPLE – Mean Deviation
Variance and Standard Deviation
The number of cappuccinos sold at the Starbucks location in the
Orange Country Airport between 4 and 7 p.m. for a sample of 5
days last year were 20, 40, 50, 60, and 80. Determine the mean
deviation for the number of cappuccinos sold.
z
z
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EXAMPLE – Population Variance and
Standard Deviation
Population Variance and Standard Deviation
z
z
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Variance
--- The arithmetic mean of the squared
deviations from the mean.
Standard Deviation
--- The square root of the variance.
The number of traffic citations issued during the last five months in
Beaufort County, South Carolina, is 38, 26, 13, 41, and 22. What
is the population variance?
Population Variance
Population standard Deviation
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EXAMPLE – Sample Variance
Sample Variance and Standard Deviation
z
z
The hourly wages for a sample of part-time employees at
Home Depot are: $12, $20, $16, $18, and $19. What is the
sample variance?
Sample Variance
Sample standard Deviation
Square root of sample variance.
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Computational Form
Chebyshev’s Theorem
n
Sum of Square (SS) =
n
∑ ( X -X ) = ∑ X
2
i
i=1
2
i
- n (X)
2
i=1
SS
Population Variance =
n
SS
Sample Variance =
n-1
≥ 1Mean -kth sd
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1
k2
Mean
Mean + kth sd
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Chebyshev’s Theorem
The Empirical Rule
The arithmetic mean biweekly amount contributed by
the Dupree Paint employees to the company’s profitsharing plan is $51.54, and the standard deviation is
$7.51. At least what percent of the contributions lie
within plus 3.5 standard deviations and minus 3.5
standard deviations of the mean?
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The Arithmetic Mean of Grouped Data Example
The Arithmetic Mean of Grouped Data
Recall in Chapter 2, we
constructed a frequency
distribution for the vehicle
selling prices. The
information is repeated
below. Determine the
arithmetic mean vehicle
selling price.
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The Arithmetic Mean of Grouped Data Example
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Standard Deviation of Grouped Data
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Standard Deviation of Grouped Data Example
Exercises
z
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1,3,7,9,13,19, 25, 27, 29, 31, 35, 39, 41, 47, 51, 53,
55, 59,61, 65, 79, 81, 83
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End of Chapter 3
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