Statistics

GOALS Understand why we study statistics. Explain what is meant by descriptive statistics and inferential statistics. Distinguish between a qualitative variable and a quantitative variable. Describe how a discrete variable is different from a continuous variable. Scales of measurement.

What is Meant by Statistics? Statistics is the science of collecting, organizing, presenting, analyzing, and interpreting numerical data to assist in making more effective decisions.

Who Uses Statistics? Statistical techniques are used extensively in marketing, accounting, quality control, consumers, professional sports people, hospital administrators, educators, politicians, physicians, etc...

Types of Statistics – Descriptive Statistics Descriptive Statistics: Methods of organizing, summarizing, and presenting data in an informative way. EXAMPLE 1: A Gallup poll found that 49% of the people in a survey knew the name of the first book of the Bible. The statistic 49 describes the number out of every 100 persons who knew the answer. EXAMPLE 2: According to Consumer Reports, General Electric washing machine owners reported 9 problems per 100 machines during 2001. The statistic 9 describes the number of problems out of every 100 machines. Inferential Statistics: A decision, estimate, prediction, or generalization about a population, based on a sample .

Population versus Sample A population is a collection of all possible individuals, objects, or measurements of interest. A sample is a portion , or part, of the population of interest

Types of Variables A. Qualitative or Attribute variable - The characteristic being studied is nonnumeric. Examples: Gender, religious affiliation, type of automobile owned, state of birth, eye color are examples. B. Quantitative variable - Information is reported numerically. Examples: Balance in your checking account, minutes remaining in class, or number of children in a family.

Quantitative Variables - Classifications Quantitative variables can be classified as either discrete or continuous . A. Discrete variables: can only assume certain values and there are usually “gaps” between values. Examples: the number of bedrooms in a house, or the number of hammers sold at the local Home Depot (1,2,3,…,etc). B. Continuous variable can assume any value within a specified range. Examples: The pressure in a tire, the weight of a pork chop, or the height of students in a class.

Four Levels of Measurement Nominal level - data that is classified into categories and cannot be arranged in any particular order. EXAMPLES: eye color, gender, religious affiliation. Ordinal level – involves data arranged in some order, but the differences between data values cannot be determined or are meaningless. EXAMPLE: During a taste test of 4 soft drinks, Mellow Yellow was ranked number 1, Sprite number 2, Seven-up number 3, and Orange Crush number 4. Interval level - similar to the ordinal level, with the additional property that meaningful amounts of differences between data values can be determined. There is no natural zero point. EXAMPLE: Temperature on the Fahrenheit scale. Ratio level - the interval level with an inherent zero starting point. Differences and ratios are meaningful for this level of measurement. EXAMPLES: Monthly income of surgeons, or distance traveled by manufacturer’s representatives per month.

Summary of the Characteristics for Levels of Measurement

Describing Data: Frequency Tables, Frequency Distributions, and Graphic Presentation

GOALS Organize qualitative data into a frequency table. Present a frequency table as a bar chart or a pie chart. Organize quantitative data into a frequency distribution. Present a frequency distribution for quantitative data using histograms, frequency polygons, and cumulative frequency polygons.

Frequency Distribution A Frequency distribution is a grouping of data into mutually exclusive categories showing the number of observations in each class.

Relative Class Frequencies Class frequencies can be converted to relative class frequencies to show the fraction of the total number of observations in each class. A relative frequency captures the relationship between a class total and the total number of observations .

Frequency Distribution Class midpoint: A point that divides a class into two equal parts. This is the average of the upper and lower class limits. Class frequency: The number of observations in each class. Class interval: The class interval is obtained by subtracting the lower limit of a class from the lower limit of the next class.

EXAMPLE – Creating a Frequency Distribution Table Ms. Kathryn Ball of AutoUSA wants to develop tables, charts, and graphs to show the typical selling price on various dealer lots. The table on the right reports only the price of the 80 vehicles sold last month at Whitner Autoplex.

Constructing a Frequency Table - Example Step 1: Decide on the number of classes. A useful recipe to determine the number of classes ( k ) is the “2 to the k rule.” such that 2 k > n . There were 80 vehicles sold. So n = 80. If we try k = 6, which means we would use 6 classes, then 2 6 = 64, somewhat less than 80. Hence, 6 is not enough classes. If we let k = 7, then 2 7 = 128, which is greater than 80. So the recommended number of classes is 7. Step 2: Determine the class interval or width. The formula is: i  (H-L)/k where i is the class interval, H is the highest observed value, L is the lowest observed value, and k is the number of classes. ($35,925 - $15,546)/7 = $2,911 Round up to some convenient number, such as a multiple of 10 or 100. Use a class width of $3,000

Step 3: Set the individual class limits Constructing a Frequency Table - Example

Step 4: Tally the vehicle selling prices into the classes. Step 5: Count the number of items in each class. Constructing a Frequency Table

Relative Frequency Distribution To convert a frequency distribution to a relative frequency distribution, each of the class frequencies is divided by the total number of observations.

Graphic Presentation of a Frequency Distribution The three commonly used graphic forms are: Histograms Frequency polygons Cumulative frequency distributions

Histogram Histogram for a frequency distribution based on quantitative data is very similar to the bar chart showing the distribution of qualitative data. The classes are marked on the horizontal axis and the class frequencies on the vertical axis. The class frequencies are represented by the heights of the bars.

Frequency Polygon A frequency polygon also shows the shape of a distribution and is similar to a histogram. It consists of line segments connecting the points formed by the intersections of the class midpoints and the class frequencies.

Cumulative Frequency Distribution

Describing Data: Numerical Measures Chapter 3

GOALS Calculate the arithmetic mean, weighted mean, median, mode, and geometric mean. Explain the characteristics, uses, advantages, and disadvantages of each measure of location. Identify the position of the mean, median, and mode for both symmetric and skewed distributions. Compute and interpret the range, mean deviation, variance, and standard deviation. Understand the characteristics, uses, advantages, and disadvantages of each measure of dispersion. Understand Chebyshev’s theorem and the Empirical Rule as they relate to a set of observations.

Characteristics of the Mean The arithmetic mean is the most widely used measure of location. It requires the interval scale. Its major characteristics are: 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.

Population Mean For ungrouped data, the population mean is the sum of all the population values divided by the total number of population values:

Sample Mean For ungrouped data, the sample mean is the sum of all the sample values divided by the number of sample values:

Properties of the Arithmetic Mean 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.

Weighted Mean The weighted mean of a set of numbers X 1 , X 2 , ..., X n , with corresponding weights w 1 , w 2 , ..., w n , is computed from the following formula:

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 Median The Median is the midpoint of the values after they have been ordered from the smallest to the largest. There are as many values above the median as below it in the data array. For an even set of values, the median will be the arithmetic average of the two middle numbers.

Properties of the Median There is a unique median for each data set. It is not affected by extremely large or small values and is therefore a valuable measure of central tendency when such values occur. It can be computed for ratio-level, interval-level, and ordinal-level data. It can be computed for an open-ended frequency distribution if the median does not lie in an open-ended class.

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. The heights of four basketball players, in inches, are: 76, 73, 80, 75 Arranging the data in ascending order gives: 73, 75, 76, 80 . Thus the median is 75.5

The Mode The mode is the value of the observation that appears most frequently.

Mean, Median, Mode Using Excel Table 2–4 in Chapter 2 shows the prices of the 80 vehicles sold last month at Whitner Autoplex in Raytown, Missouri. Determine the mean and the median selling price. The mean and the median selling prices are reported in the following Excel output. There are 80 vehicles in the study. So the calculations with a calculator would be tedious and prone to error.

Mean, Median, Mode Using Excel

The Relative Positions of the Mean, Median and the Mode

The Geometric Mean 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. The geometric mean of a set of n positive numbers is defined as the n th root of the product of n values. The formula for the geometric mean is written:

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. Why is this so? We begin by calculating the geometric mean.

EXAMPLE – Geometric Mean (2) 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?

Dispersion Why Study Dispersion? 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. 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. A second reason for studying the dispersion in a set of data is to compare the spread in two or more distributions.

Measures of Dispersion Range Mean Deviation Variance and Standard Deviation

EXAMPLE – Range 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. Range = Largest – Smallest value = 80 – 20 = 60

EXAMPLE – Mean 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.

EXAMPLE – Variance and Standard Deviation 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?

EXAMPLE – Sample Variance 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?

Chebyshev’s Theorem The arithmetic mean biweekly amount contributed by the Dupree Paint employees to the company’s profit-sharing 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?

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. The Arithmetic Mean of Grouped Data - Example

The Arithmetic Mean of Grouped Data - Example

Standard Deviation of Grouped Data

Standard Deviation of Grouped Data - Example Refer to the frequency distribution for the Whitner Autoplex data used earlier. Compute the standard deviation of the vehicle selling prices

Describing Data: Displaying and Exploring Data Chapter 4

Develop and interpret a dot plot. Develop and interpret a stem-and-leaf display. Compute and understand quartiles, deciles, and percentiles. Construct and interpret box plots. Compute and understand the coefficient of skewness. Draw and interpret a scatter diagram. Construct and interpret a contingency table. GOALS

Dot Plots A dot plot groups the data as little as possible and the identity of an individual observation is not lost. To develop a dot plot, each observation is simply displayed as a dot along a horizontal number line indicating the possible values of the data. If there are identical observations or the observations are too close to be shown individually, the dots are “piled” on top of each other.

Dot Plots - Examples Reported below are the number of vehicles sold in the last 24 months at Smith Ford Mercury Jeep, Inc., in Kane, Pennsylvania, and Brophy Honda Volkswagen in Greenville, Ohio. Construct dot plots and report summary statistics for the two small-town Auto USA lots.

Stem-and-Leaf In Chapter 2, we showed how to organize data into a frequency distribution. The major advantage to organizing the data into a frequency distribution is that we get a quick visual picture of the shape of the distribution. One technique that is used to display quantitative information in a condensed form is the stem-and-leaf display. Stem-and-leaf display is a statistical technique to present a set of data. Each numerical value is divided into two parts. The leading digit(s) becomes the stem and the trailing digit the leaf. The stems are located along the vertical axis, and the leaf values are stacked against each other along the horizontal axis. Advantage of the stem-and-leaf display over a frequency distribution - the identity of each observation is not lost.

Stem-and-Leaf – Example Suppose the seven observations in the 90 up to 100 class are: 96, 94, 93, 94, 95, 96, and 97 . The stem value is the leading digit or digits, in this case 9. The leaves are the trailing digits. The stem is placed to the left of a vertical line and the leaf values to the right. The values in the 90 up to 100 class would appear as Then, we sort the values within each stem from smallest to largest. Thus, the second row of the stem-and-leaf display would appear as follows:

Stem-and-leaf: Another Example Listed in Table 4–1 is the number of 30-second radio advertising spots purchased by each of the 45 members of the Greater Buffalo Automobile Dealers Association last year. Organize the data into a stem-and-leaf display. Around what values do the number of advertising spots tend to cluster? What is the fewest number of spots purchased by a dealer? The largest number purchased?

Stem-and-leaf: Another Example

Stem-and-leaf: Another Example (Minitab)

The standard deviation is the most widely used measure of dispersion. Alternative ways of describing spread of data include determining the location of values that divide a set of observations into equal parts. These measures include quartiles, deciles, and percentiles. Quartiles, Deciles and Percentiles

To formalize the computational procedure, let L p refer to the location of a desired percentile. So if we wanted to find the 33rd percentile we would use L 33 and if we wanted the median, the 50th percentile, then L 50 . The number of observations is n, so if we want to locate the median, its position is at ( n + 1)/2, or we could write this as ( n + 1)( P /100), where P is the desired percentile. Percentile Computation

Percentiles - Example Listed below are the commissions earned last month by a sample of 15 brokers at Salomon Smith Barney’s Oakland, California, office. Salomon Smith Barney is an investment company with offices located throughout the United States. $2,038 $1,758 $1,721 $1,637 $2,097 $2,047 $2,205 $1,787 $2,287 $1,940 $2,311 $2,054 $2,406 $1,471 $1,460 Locate the median, the first quartile, and the third quartile for the commissions earned.

Percentiles – Example (cont.) Step 1: Organize the data from lowest to largest value $1,460 $1,471 $1,637 $1,721 $1,758 $1,787 $1,940 $2,038 $2,047 $2,054 $2,097 $2,205 $2,287 $2,311 $2,406

Percentiles – Example (cont.) Step 2: Compute the first and third quartiles. Locate L 25 and L 75 using:

Percentiles – Example (Minitab)

Percentiles – Example (Excel)

Boxplot – Using Minitab Refer to the Whitner Autoplex data in Table 2–4. Develop a box plot of the data. What can we conclude about the distribution of the vehicle selling prices?

Skewness In Chapter 3, measures of central location for a set of observations (the mean, median, and mode) and measures of data dispersion (e.g. range and the standard deviation) were introduced Another characteristic of a set of data is the shape . There are four shapes commonly observed: symmetric, positively skewed, negatively skewed, bimodal.

Skewness - Formulas for Computing The coefficient of skewness can range from -3 up to 3. A value near -3, such as -2.57, indicates considerable negative skewness. A value such as 1.63 indicates moderate positive skewness. A value of 0, which will occur when the mean and median are equal, indicates the distribution is symmetrical and that there is no skewness present.

Skewness – An Example Following are the earnings per share for a sample of 15 software companies for the year 2005. The earnings per share are arranged from smallest to largest. Compute the mean, median, and standard deviation. Find the coefficient of skewness using Pearson’s estimate. What is your conclusion regarding the shape of the distribution?

Skewness – An Example Using Pearson’s Coefficient

Skewness – A Minitab Example

Describing Relationship between Two Variables One graphical technique we use to show the relationship between variables is called a scatter diagram. To draw a scatter diagram we need two variables. We scale one variable along the horizontal axis ( X -axis) of a graph and the other variable along the vertical axis ( Y -axis).

Describing Relationship between Two Variables – Scatter Diagram Examples

In the Introduction to Chapter 2 we presented data from AutoUSA. In this case the information concerned the prices of 80 vehicles sold last month at the Whitner Autoplex lot in Raytown, Missouri. The data shown include the selling price of the vehicle as well as the age of the purchaser. Is there a relationship between the selling price of a vehicle and the age of the purchaser? Would it be reasonable to conclude that the more expensive vehicles are purchased by older buyers? Describing Relationship between Two Variables – Scatter Diagram Excel Example

Describing Relationship between Two Variables – Scatter Diagram Excel Example

Contingency Tables A scatter diagram requires that both of the variables be at least interval scale . What if we wish to study the relationship between two variables when one or both are nominal or ordinal scale ? In this case we tally the results in a contingency table.

Contingency Tables – An Example A manufacturer of preassembled windows produced 50 windows yesterday. This morning the quality assurance inspector reviewed each window for all quality aspects. Each was classified as acceptable or unacceptable and by the shift on which it was produced. Thus we reported two variables on a single item. The two variables are shift and quality. The results are reported in the following table.

Statistics

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Statistics