8.1 Measures of Central Tendency: 8.1 Measures of Central Tendency Find the mean, median, mode, and range for a set of data Construct a frequency distribution table and calculate the mean, median, mode, and range


Measures of Central Tendency: Measures of Central Tendency In the study of statistics there are three types of averages called the mean, median, and mode. As a group, these averages are called measures of central tendency.


Mean: Mean The mean, or arithmetic average, is found by adding a group of numbers and dividing the sum by the number of items added. The mean is the best known and most used measure of central tendency. The group of numbers is sometimes referred to as the data or data set.


Median: Median The median is the middle number in a set of data that is arranged in either ascending or descending order. One-half of the numbers will be on either side of the median. To find the median, arrange the numbers in order. If the group has an odd count of numbers, the number in the middle is the median. If the group has an even count of numbers, add the two middle numbers and divide by 2 to find the median.


Mode: Mode The mode is the number that occurs most frequently in a group of numbers arranged in order. There may be no mode or more than one mode in a set of data.


Range: Range Often when you work with measures of central tendency, you also find the range, which is not a measure of central tendency, rather a measure of dispersion. The range is the difference between the highest and lowest numbers in a set of data. Highest Number – Lowest Number = Range


Frequency Distribution: Frequency Distribution When you work with larger sets of data, it may be difficult to list them in order. Making a frequency distribution is one way of arranging numbers. This method called tallying makes counting easier. From the frequency distribution table, you can calculate the mean, median, mode, and range.


8.2 Probability: 8.2 Probability Compute the probability of simple events Compute experimental probability Compute probability based on experience


Probability: Probability Probability is a way of mathematically predicting the chance an event will occur.


Chance Events: Chance Events You have two marbles that are alike in every way except color. The marbles, one blue and one white, are placed in a bag. Suppose you are asked to pick one from the bag without looking. You have no way of knowing which marble is blue and which marble is white. This means that the marble you pick will be a random choice, and the outcome will be a chance event.


Probability of a Chance Event: Probability of a Chance Event With this bag of marbles, what are the chances that your outcome will be to pick the blue marble? Two outcomes are possible: 1) you pick the blue marble or 2) you pick the white marble. Since your pick is a random choice, one outcome is just as likely as the other. The outcomes are equally likely. The chance that you will pick the blue marble is 1 out of 2, shown as ½. You may also say that the probability is 0.50, or 50%.


Certain or Impossible Events: Certain or Impossible Events If a bag holds nothing but 5 yellow marbles, the chances of picking a yellow marble are 5 out of 5, which is 5/5 or 1. In this case picking a yellow marble is an event that is certain to happen—no other outcome is possible. Any event that is certain has a probability of 1. From a bag holding nothing but yellow marbles, the event that you will pick a white marble is an impossible outcome. The chances of picking a white marble are 0 out of 5, which is 0/5, or 0. Any event that is impossible has a probability of 0.


Probability and Large Number: Probability and Large Number The probability of an event occurring tells you about how many times you can expect the event to happen in a large number of tries. For example, if a perfectly balanced coin is flipped, it is as likely to come up heads as it is tails. So, you can say that the probability of heads is ½. This does not mean that out of 40 flips you are sure to get 20 heads. You may get several more or several less than 20. (continued on the next slide)


Probability and Large Number (continued): Probability and Large Number (continued) What it does mean is if you flip the coin many times, say 1,000 times, the number of heads should not be far from ½ of 1,000. The longer you keep flipping, the nearer the number of heads should come to ½ the number of flips. This is called the principle of large numbers. It is one of the main concepts of probability. For a large number of tries of the same kind, you can predict the outcome with only a small amount of error.


Experimental Probability: Experimental Probability In the examples so far you could calculate the probability of the event from the description of the conditions. In many cases, this information may not be known, and the probability must be based on experiment. The quality control departments of companies do not test every product they make. They often test a few of the products to see how many are up to standard and how many are not. (continued on the next slide)


Experimental Probability (continued): Experimental Probability (continued) If these samples are chosen by chance, they are called random samples. Suppose 100 staplers were randomly selected and tested and 3 were found to be defective. The probability of finding a defective stapler is 3/100, 0.03, or 3%. On the basis of this test, you could estimate that about 3% of the whole batch of staplers from which the samples were taken will be defective.


Probability Based on Experience: Probability Based on Experience Life insurance companies use records of births and deaths to make mortality tables that show how many persons in a sample group of 100,000 live babies have reached certain ages.


Mortality Table: Mortality Table From the mortality table, you can calculate the probability that a person will reach a particular age. For example, 93,000 of the sample of 100,000 people born reached the age of 40. So at birth, the probability of living to be 40 is 93,000 ÷ 100,000 or 93%.


8.3 Bar and Line Graphs: 8.3 Bar and Line Graphs Interpret and make vertical bar graphs Interpret and make horizontal bar graphs Interpret and make line graphs


Graphs: Graphs Business firms use graphs to show data about their companies or industries. Graphs often show facts and trends more clearly than do numbers in tables.


Vertical Bar Graph: Vertical Bar Graph The vertical bar graph shown at the top of page 358 (with bars running up and down) shows the daily sales of The Building Center for a week. There is a bar for Monday through Saturday. The height of each bar shows the sales for each day. The scale for measuring the bars is on the left side of the graph. Each vertical block on the graph equals $100 of sales. (continued on the next slide)


Vertical Bar Graph (continued): Vertical Bar Graph (continued) The lines for $500 and multiples of $500 are labeled. The height of each bar is to the nearest $50. The heading or title of the graph shows the company name, identifies the data shown, and gives the time period that the graph represents.


Sample Vertical Bar Graph: Sample Vertical Bar Graph


Horizontal Bar Graph: Horizontal Bar Graph The horizontal bar graph at the top page 359 (with bars running left to right) shows the sales by department of The Building Center. It looks like the vertical bar graph except that the bars are horizontal. Each horizontal block on the graph equals $2,000. The amounts for each bar were rounded to the nearest $1,000 before the graph was made.


Sample Horizontal Bar Graph: Sample Horizontal Bar Graph


Line Graph: Line Graph The line graph shown at the top of page 600 shows the sales of The Building Center by months. The time scale runs from left to right and is at the bottom of the graph. The dollar scale runs from bottom to top at the left. The monthly sales were rounded to the nearest $1,000. The line was made by first placing dots showing each month’s sales. The dots were then connected by drawing a line.


Sample Line Graph: Sample Line Graph


8.4 Circle and Rectangle Graphs: 8.4 Circle and Rectangle Graphs Make circle graphs Interpret and make rectangle graphs


Circle Graphs: Circle Graphs Bar graphs are frequently used to compare quantities to each other. Line graphs show change over time. Circle graphs are used to show how parts relate to the whole and to each other. Circle graphs are based on a full circle of 360 degrees (360°), which is the whole, or 100%. The circle graph is divided into parts, called sectors.


Sample Circle Graph: Sample Circle Graph


Rectangle Graphs: Rectangle Graphs A rectangle graph is a single vertical or horizontal rectangular bar that is divided into sections. The entire rectangle represents the whole, or 100%. The sections show the parts of the whole. A rectangle graph shows how the parts relate to each other and to the whole. Each part is proportional in size to the whole. Rectangle graphs are most often used to show dollars or percents.


Vertical Rectangle Graph: Vertical Rectangle Graph In the vertical rectangle graph shown at the right, the whole rectangle represents New Solution’s sales of $2,000,000 for a month. Parts of the rectangle, also shown in dollars, represent each division’s contribution to the monthly sales.


Horizontal Rectangle Graph: Horizontal Rectangle Graph The horizontal rectangle graph shown above displays the same information as the vertical rectangle graph. However, the data is presented as percents.


8.5 Economic Statistics: 8.5 Economic Statistics Interpret consumer price index data Calculate rates of inflation Calculate the purchasing power of the dollar Analyze unemployment data


Consumer Price Index: Consumer Price Index The Consumer Price Index (CPI) is a widely reported measure of how much the prices of goods and services typically bought by consumers have changed when compared to a base period. A base period is a period of time with which comparisons are made. The base period for most of the items in the CPI is the 1982-84 period.


Index Number: Index Number The CPI uses a single number, called an index number, to compare price changes to the base period. The index number for the base period is always equal to 100.


Historical Report: Historical Report The Historical Report of the CPI shows index numbers for various categories of consumer goods and services. The All Items column gives an average number considering all categories and is the number commonly used when referring to the CPI. Note that two categories, Recreation and Education & Communication, were added to the CPI in 1997, so 1997 is their base period.


Sample Historical Report: Sample Historical Report


Expression of CPI: Expression of CPI The CPI may be expressed in several ways. For example, the CPI for “All Items” in 2003 is 184.0. This means that the cost of goods in 2003 was 184.0% of their cost in the base period. The percent increase in prices from the base period to 2003 is 84.0% (184.0 – 100.0). Looking at the relationship in another way, it cost $184 in 2003 to buy the same goods for which you would have paid $100 in the base period.


Rate of Inflation: Rate of Inflation For consumers, business firms, organizations, and the government, inflation means that the prices of goods and services they buy are rising. The U.S. Department of Labor publishes the Consumer Price Index report that tells how much inflation has occurred within the past year. A calculation in the report, called the rate of inflation, shows the percent increase in prices from the previous year.


Purchasing Power of the Dollar: Purchasing Power of the Dollar When inflation occurs, each dollar buys less than it did in the past. The purchasing power of the dollar is a measure of how much a dollar now buys compared to what it could buy during some base period. The base period is a time period with which all purchasing power of the dollar comparisons are made.


Unemployment Rate: Unemployment Rate The unemployment rate tells the percentage of the total labor force that is not working. The labor force consists of all people who are willing to work and who either have a job or are looking for a job. The table below shows the unemployment rate for different persons for one month as estimated by the U.S. Department of Labor.


Sample Unemployment Information: Sample Unemployment Information