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发布时间：2014-01-05 14:00:47

CHAPTER 3: Statistical Description of Data

to accompany

Introduction to Business Statistics

fourth edition, by Ronald M. Weiers

Presentation by Priscilla Chaffe-Stengel Donald N. Stengel

? 2002 The Wadsworth Group

Chapter 3 - Learning Objectives

? Describe data using measures of central tendency and dispersion:

– for a set of individual data values, and – for a set of grouped data.

? Convert data to standardized values. ? Use the computer to visually represent data. ? Use the coefficient of correlation to measure association between two quantitative variables.

? 2002 The Wadsworth Group

Chapter 3 - Key Terms

? Measures of Central Tendency, ? Mean

– μ, population; x , sample

The Center

? Weighted Mean ? Median ? Mode (Note comparison of mean, median, and mode)

? 2002 The Wadsworth Group

Chapter 3 - Key Terms

? Measures of Dispersion, ? Range ? Mean absolute deviation ? Variance

(Note the computational difference between s2 and s2.)

The Spread

? ? ? ?

Standard deviation Interquartile range Interquartile deviation Coefficient of variation

? 2002 The Wadsworth Group

Chapter 3 - Key Terms

? Measures of Relative Position ? Quantiles

– Quartiles – Deciles – Percentiles

? Residuals ? Standardized values

? 2002 The Wadsworth Group

Chapter 3 - Key Terms

? Measures of Association ? Coefficient of correlation, r

– Direction of the relationship: direct (r > 0) or inverse (r < 0) – Strength of the relationship: When r is close to 1 or –1, the linear relationship between x and y is strong. When r is close to 0, the linear relationship between x and y is weak. When r = 0, there is no linear relationship between x and y.

? Coefficient of determination, r2

– The percent of total variation in y that is explained by variation in x.

? 2002 The Wadsworth Group

? Mean

The Center: Mean

?Population: μ= (Sxi)/N ?Sample: = (Sxi)/n

– Arithmetic average = (sum all values)/# of values Be sure you know how to get the value easily from your calculator and computer softwares.

Problem: Calculate the average number of truck shipments from the United States to five Canadian cities for the following data given in thousands of bags: Montreal, 64.0; Ottawa, 15.0; Toronto, 285.0; Vancouver, 228.0; Winnipeg, 45.0 (Ans: 127.4)

? 2002 The Wadsworth Group

x

The Center: Weighted Mean

? When what you have is grouped data, compute the mean using μ= (Swixi)/Swi

Problem: Calculate the average profit from truck shipments, United States to Canada, for the following data given in thousands of bags and profits per thousand bags: Montreal 64.0 Ottawa 15.0 Toronto 285.0 $15.00 $13.50 $15.50 Vancouver 228.0 Winnipeg 45.0 $12.00 $14.00

(Ans: $14.04 per thous. bags)

? 2002 The Wadsworth Group

? To find the median:

The Center: Median

1. Put the data in an array. 2A. If the data set has an ODD number of numbers, the median is the middle value. 2B. If the data set has an EVEN number of

numbers, the median is the AVERAGE of the middle two values. (Note that the median of an even set of data values is not necessarily a member of the set of values.)

? The median is particularly useful if there are outliers in the data set, which otherwise tend to sway the value of an arithmetic mean.

? 2002 The Wadsworth Group

The Center: Mode

? The mode is the most frequent value. ? While there is just one value for the mean and one value for the median, there may be more than one value for the mode of a data set. ? The mode tends to be less frequently used than the mean or the median.

? 2002 The Wadsworth Group

Comparing Measures of Central Tendency

? If mean = median = mode, the shape of the distribution is symmetric. ? If mode < median < mean or if mean > median > mode, the shape of the distribution trails to the right, is positively skewed. ? If mean < median < mode or if mode > median > mean, the shape of the distribution trails to the left, is negatively skewed.

? 2002 The Wadsworth Group

The Spread: Range

? The range is the distance between the smallest and the largest data value in the set. ? Range = largest value – smallest value ? Sometimes range is reported as an interval, anchored between the smallest and largest data value, rather than the actual width of that interval.

? 2002 The Wadsworth Group

Key Concept - Residuals

? Residuals are the differences between each data value in the set and the group mean:

– for a population, xi – μ – for a sample, xi – x

? 2002 The Wadsworth Group

The Spread: MAD

? The mean absolute deviation is found by summing the absolute values of all residuals and dividing by the number of values in the set: for a population, MAD = (S|xi – μ|)/N for a sample, MAD = (S|xi – x |)/n

? 2002 The Wadsworth Group

The Spread: Variance

? Variance is one of the most frequently used measures of spread,

S(x –?)2 S(x )2 – N ??2 – for population, s 2 ? i i ? N N

– for sample,

? The right side of each equation is often used as a computational shortcut.

? 2002 The Wadsworth Group

S(x – x )2 S(x )2 – n?x 2 i i s2 ? ? n –1 n–1

The Spread: Standard Deviation

? Since variance is given in squared units, we often find uses for the standard deviation, which is the square root of variance: – for a population, s ? s 2

– for a sample, s ? s2 Be sure you know how to get the values easily from your calculator and computer softwares.

? 2002 The Wadsworth Group

Coefficient of Variation

? The coefficient of variation (CV) expresses the standard deviation as a percent of the mean, indicating the relative amount of dispersion in the data. CV ? s ?100% ?

? 2002 The Wadsworth Group

Relative Position - Quartiles

? One of the most frequently used quantiles is the quartile. ? Quartiles divide the values of a data set into four subsets of equal size, each comprising 25% of the observations. ? To find the first, second,

and third quartiles:

– – – – 1. Arrange the N data values into an array. 2. First quartile, Q1 = data value at position (N + 1)/4 3. Second quartile, Q2 = data value at position 2(N + 1)/4 4. Third quartile, Q3 = data value at position 3(N + 1)/4

? 2002 The Wadsworth Group

What is a Standardized Value?

? How far above or below the individual value is compared to the population mean in units of standard deviation

– “How far above or below”= (data value – mean) which is the residual... – “In units of standard deviation” = divided by s

? Standardized data value

x –? ? z

– A negative z means the data value falls below the mean.

? 2002 The Wadsworth Group

s

Why is a Standardized Value Important?

? Chebyshev’s Theorem: For either a sample or a population, the percentage of observations that fall within k (for k > 1) standard deviations of the mean will be at least

(1– 1 )?100% k2

? 2002 The Wadsworth Group

Why is a Standardized Value Important?

? The Empirical Rule: For bell-shaped, symmetric distributions,

– about 68% of the observations will fall within 1 standard deviation of the mean, – about 95% of the observations will fall within 2 standard deviations of the mean, – practically all of the observations will fall within 3 standard deviations of the mean.

? 2002 The Wadsworth Group

An Example: Problem 3.54

A law enforcement agency administering breathalyzer tests to a sample of drivers stopped at a New Year’s Eve roadblock measured the following blood alcohol levels for the 25 drivers who were stopped:

0.00% 0.04% 0.05 % 0.00 % 0.03 % 0.08% 0.00 % 0.21 % 0.09 % 0.00 % 0.15% 0.03 % 0.01 % 0.05 % 0.16 % 0.18% 0.11 % 0.10 % 0.03 % 0.04 % 0.02% 0.17% 0.19 % 0.00 % 0.10 %

? 2002 The Wadsworth Group

Problem 3.54, continued

? Calculate the mean and standard deviation from this sample.

Ans:

Mean = 0.0736% Standard Deviation = 0.0684%

? 2002 The Wadsworth Group

Problem 3.54, continued

? Use Chebyshev’s Theorem to determine the minimum percentage of observations that should fall within k = 1.50 units of standard deviation from the mean.

1 )?100%?(1– 1 )?100% Ans: (1– 2 2 k 1.50 ?(1– 0.4444)?100%?55.55%

At least 55.55% of the data values should fall within k = 1.50 units of standard deviation from the mean.

? 2002 The Wadsworth Group

Problem 3.54, continued

? Do the sample results support Chebyshev’s Theorem?

Ans: 1.50 (s) = 0.1026% mean + 1.50 (s) = 0.0736% + 0.1026% = 0.1762% mean – 1.50 (s) = 0.0736% – 0.1026% = – 0.0290% A total of 22/25 data values fall in this interval, or 88% of the sample. Yes, the data support Chebyshev’s Theorem.

? 2002 The Wadsworth Group

Problem 3.54, continued

? Calculate the coefficient of variation for these data. Ans: s ?100%? 0.0684%?100%?92.9% CV ? ? 0.0736%

? 2002 The Wadsworth Group