Business Statistics Chapter 1

Explain what is meant by statistics 1-1 GOALS: Explain what is meant by statistics Explain what is meant by descriptive statistics and inferential statistics. Distinguish between a qualitative variable and a quantitative variable; discrete variable and a continuous variable. Define the terms mutually exclusive and exhaustive. Distinguish among the nominal, ordinal, interval, and ratio levels of measurement.

What is meant by Statistics? 1-2 What is meant by Statistics? Statistics is the science of data which involves collecting, classifying summarizing, organizing, analyzing, and interpreting numerical information to assist in making effective decision

Descriptive Statistics Inferential Statistics Types of Statistics Descriptive Statistics Inferential Statistics Descriptive Statistics: Methods of organizing, summarizing, and presenting data in an informative way. Inferential Statistics: A decision, estimate, prediction, or generalization about a population, based on a sample.

Types of Statistics Inferential Statistics: Descriptive Statistics: Methods of organizing, summarizing, and presenting data in an informative way. Inferential Statistics: A decision, estimate, prediction, or generalization about a population, based on a sample.

Types of Statistics (examples of inferential statistics) Eg 1: TV networks constantly monitor the popularity of heir programs by hiring Nielsen and other organizations to sample the preferences of TV viewers. Eg 2: The accounting department of a large firm will select a sample of the invoices to check for accuracy for all the invoices of the company. Eg 3: Wine tasters sip a few drops of wine to make a decision with respect to all the wine waiting to be released for sale.

Types of Statistics A population is a collection of all possible individuals, objects, or measurements of interest. A parameter is a descriptive measure of the entire population of all observations of interest A sample is a portion, or part, of the population of interest. A statistic describes a sample and serves as an estimate of the corresponding population parameter

Types of Variables DATA Qualitative or attribute (type of car owned) Quantitative or Numerical Discrete (number of children) Continuous (time taken for an exam)

Types of Variables For a Qualitative or Attribute variable the characteristic being studied is nonnumeric. Gender, religious affiliation, type of automobile owned, state of birth, eye color are examples. In a Quantitative variable information is reported numerically. balance in your checking account, minutes remaining in class, or number of children in a family. Quantitative variables can be classified as either discrete or continuous.

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

Sources of Data Primary data : Collected for specific purpose Direct Observation Questionnaires Interviewing Secondary Data : Collected for another purpose

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

Sources of Statistical Data For Researching problems usually requires published data. Statistics on these problems can be found in published articles, journals, and magazines. Published data is not always available on a given subject. In such cases, information will have to be collected and analyzed. One way of collecting data is via questionnaires. What are the other data collection methods?

Frequency Distributions and Graphic Presentation Chapter Two Describing Data: Frequency Distributions and Graphic Presentation

Presentation of Data Row data reveals very little Shows how a large data set can be organized and managed to provide a quick visual interpretation of the massage the data convey. Methods of Data Presentation Data Array Tabulation of Data Stem-and-Leaf display Frequency Distribution

Arrange data in systematic way Data Array Arrange data in systematic way (Ascending data array & Descending data Array) Tabulation of Data Arrange data in a tables (Rows & Columns) 1st Year 2nd Year Physical 320 160 Bio 246 126

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

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.

Organize the data into a frequency distribution. Eg 1: Dr. Tillman is Dean of the School of Business Socastee University. He wishes prepare to a report showing the number of hours per week students spend studying. He selects a random sample of 30 students and determines the number of hours each student studied last week. 15.0, 23.7, 19.7, 15.4, 18.3, 23.0, 14.2, 20.8, 13.5, 20.7, 17.4, 18.6, 12.9, 20.3, 13.7, 21.4, 18.3, 29.8, 17.1, 18.9, 10.3, 26.1, 15.7, 14.0, 17.8, 33.8, 23.2, 12.9, 27.1, 16.6. Organize the data into a frequency distribution.

There are 30 observations Two raised to the fifth power is 32. Eg 1: continued There are 30 observations Two raised to the fifth power is 32. Therefore, we should have at least 5 classes. It turns out we will need 6. The range is 23.5 hrs, found by 33.8 hrs – 10.3 hrs. We choose an interval of 5 hrs. The lower limit of the first class is 7.5 hrs.26

Suggestions on Constructing a Frequency Distribution The class intervals used in the frequency distribution should be equal. Determine a suggested class interval by using the formula: Use the computed suggested class interval to construct the frequency distribution.

Suggestions on Constructing a Frequency Distribution Note: this is a suggested class interval; if the computed class interval is 97, it may be better to use 100. Count the number of values in each class. Eg 1: A relative frequency distribution shows the percent of observations in each class.

EXAMPLE Mr. Jayatissa wishes prepare to a report showing the number of hours per week students spend studying. He selects a random sample of 30 students and determines the number of hours each student studied last week. 15.0, 23.7, 19.7, 15.4, 18.3, 23.0, 14.2, 20.8, 13.5, 20.7, 17.4, 18.6, 12.9, 20.3, 13.7, 21.4, 18.3, 29.8, 17.1, 18.9, 10.3, 26.1, 15.7, 14.0, 17.8, 33.8, 23.2, 12.9, 27.1, 16.6. Organize the data into a frequency distribution.

Relative Frequency Distribution

Graphic Presentation of a Frequency Distribution The three commonly used graphic forms are histograms, frequency polygons, and a cumulative frequency distribution. A Histogram is a graph in which 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 and the bars are drawn adjacent to each other.

Graphic Presentation of a Frequency Distribution A frequency polygon consists of line segments connecting the points formed by the class midpoint and the class frequency. A cumulative frequency distribution is used to determine how many or what proportion of the data values are below or above a certain value.

Histogram for Hours Spent Studying

Frequency Polygon for Hours Spent Studying

Cumulative Frequency Distribution For Hours Studying

Bar Chart A bar chart can be used to depict any of the levels of measurement (nominal, ordinal, interval, or ratio). Eg 3: Construct a bar chart for the number of unemployed per 100,000 population for selected cities during 2001

Bar Chart for the Unemployment Data

Pie Chart A pie chart is useful for displaying a relative frequency distribution. A circle is divided proportionally to the relative frequency and portions of the circle are allocated for the different groups. Eg 4: A sample of 200 runners were asked to indicate their favorite type of running shoe. Draw a pie chart based on the following information. Type of shoes # of runners Nike 92 Adidas 49 Reebook 37 Asics 13 Other 9

Pie Chart for Running Shoes

Chapter Three Describing Data: Measures of Central Tendency

Population Mean For ungrouped data, the population mean is the sum of all the population values divided by the total number of population values: where µ is the population mean. N is the total number of observations. X is a particular value.  indicates the operation of adding.

A Parameter is a measurable characteristic of a population. Eg 1: The Kiers family owns four cars. The following is the current mileage on each of the four cars: 56,000, 23,000, 42,000, 73,000 Find the mean mileage for the cars.

Sample Mean For ungrouped data, the sample mean is the sum of all the sample values divided by the number of sample values: Where n is the total number of values in the sample.

A statistic is a measurable characteristic of a sample. Eg 2: A sample of five executives received the following bonus last year ($000): 14.0, 15.0, 17.0, 16.0, 15.0

Eg 3: Consider the set of values: 3, 8, and 4. The mean is 5 Eg 3: Consider the set of values: 3, 8, and 4. The mean is 5. Illustrating the fifth property: Weighted Mean The weighted mean of a set of numbers X1, X2, ..., Xn, with corresponding weights w1, w2, ...,wn, is computed from the following formula:

Eg 6: During a one hour period on a hot Saturday afternoon cabana boy Chris served fifty drinks. He sold five drinks for $0.50, fifteen for $0.75, fifteen for $0.90, and fifteen for $1.10. Compute the weighted mean of the price of the drinks.

The Median The Median is the midpoint of the values after they have been ordered from the smallest to the largest. There areas 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.

Eg 4 : 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. Eg 5 : 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. Eg 5: The exam scores for ten students are: 81, 93, 84, 75, 68, 87, 81, 75, 81, 87. Because the score of 81 occurs the most often, it is the mode.

Geometric Mean The geometric mean (GM) of a set of n numbers is defined as the nth root of the product of the n numbers. The formula is: The geometric mean is used to average percents, indexes, and relatives.

The interest rate on three bonds were 5, 21, and 4 percent. The geometric mean is The arithmetic mean is (5+21+4)/3 =10.0 The GM gives a more conservative profit figure because it is not heavily weighted by the rate of 21percent.

Geometric Mean continued Another use of the geometric mean is to determine the percent increase in sales, production or other business or economic series from one time period to another.

Eg 8 : The total number of females enrolled in American colleges increased from 755,000 in 1992 to 835,000 in 2000. That is, the geometric mean rate of increase is 1.27%.

Eg 9 : There are many flights from Houston to Little Rock, AK each day Eg 9 : There are many flights from Houston to Little Rock, AK each day. The data below shows the number of minutes a flight was late (or early) in arriving in Little Rock for a sample of 5 flights. To explain, a positive number means the flight was late, a value of 0 indicates it arrived on time, and a negative number indicates it was early. So the first flight was 4 minutes late and the last flight 10 minutes early. 4 12 -9 6 -10 Determine the mean amount flights were late (or early). b. Determine the median amount flights were late (or early).

Determine the mean amount flights were late (or early). b. Determine the median amount flights were late (or early).

Eg10: Suppose your cousin started a management job with Ford Motor Company in 1990 at $30,000 per year. In the year 2002 her salary was $65,000. What was the geometric mean rate of increase per year for the period? Eg 11: For a sample of 50 stocks traded yesterday on the American Stock Exchange, 10 showed a decline of $1.00, 15 showed no change, and 25 increased by $2.00. Find the weighted mean.

The Mean of Grouped Data The mean of a sample of data organized in a frequency distribution is computed by the following formula: Eg12: A sample of ten movie theaters in a large metropolitan area tallied the total number of movies showing last week. Compute the mean number of movies showing.

Eg 12 : continued

The Median of Grouped Data The median of a sample of data organized in a frequency distribution is computed by: where L is the lower limit of the median class, CF is the cumulative frequency preceding the median class, f is the frequency of the median class, and i is the median class interval.

Finding the Median Class To determine the median class for grouped data: Construct a cumulative frequency distribution. Divide the total number of data values by 2. Determine which class will contain this value. For example, if n=50, 50/2 = 25, then determine which class will contain the 25th value.

Eg13 : From the table, L=5, n=10, f=3, i=2, CF=3

The Mode of Grouped Data The mode for grouped data is approximated by the midpoint of the class with the largest class frequency. The modes in Eg 13 are 8 and 10. When two values occur a large number of times, the distribution is called bimodal, as in Eg 13.

a. Determine the mean number of students per section. Eg14: The following frequency distribution reports the number of students enrolled in each of the 50 sections of various courses taught in the College of Business last summer. Students Frequency 0 up to 10 3 10 up to 20 8 20 up to 30 16 30 up to 40 10 40 up to 50 9 50 up to 60 4 Total 50 a. Determine the mean number of students per section. b. Determine the median number of students per section.

End of the Chapter Thank you Questions