Chapter 2 Descriptive Statistics o 2.1 Frequency Distributions and Their Graphs o Frequency Distributions o Graphs of Frequency Distributions

Transcription

1 Chapter 2 Descriptive Statistics o 2.1 Frequency Distributions and Their Graphs o Frequency Distributions o Graphs of Frequency Distributions o 2.2 More Graphs and Displays o Graphing Quantitative Data Sets o Graphing Qualitative Data Sets o Graphing Paired Data Sets o 2.3 Measures of Central Tendency o Mean, Median, and Mode o Weighted Mean and Mean of Grouped Data o The Shapes of Distributions o 2.4 Measures of Variation o Range o Deviation, Variance, and Standard Deviation o Interpreting Standard Deviation o Standard Deviation for Grouped Data o 2.5 Measures of Position o Quartiles o Percentiles and Other Fractiles o The Standard Score

2 2.1 Frequency Distributions and Their Graphs Part 1 Frequency Distributions Vocabulary Center Shape Variability Frequency f Frequency Distribution Classes Intervals Lower Class Limit Upper Class Limit Class Width Range Midpoint Relative Frequency Cumulative Frequency Remember a data set is a set of qualitative or quantitative data. When organizing and describing a data set there are three important characteristics: its center, its variability (its spread), and its shape. Frequency Distribution is a table that shows the different classes or intervals that the data fit in and a count of the number of entries in each class. Frequency f - is the frequency of a class, and it is the number of data entries in that class. Lower Class Limit the least number that could be in a class. Upper Class Limit the largest number that could be in a class. Class Width the lower limits of two consecutive classes subtracted, or the upper limits of two consecutive classes subtracted.

3 Range the maximum entry minus the minimum entry. How to Construct a Frequency Distribution from a Data Set Examples Pg , Example 1 and Try it Yourself 1

4 Example 1 The following sample data set lists the prices (in dollars) of 30 portable GPS navigators. Construct a frequency distribution that has seven class. Solution: Following the guidelines. 1.) The problem states there should be 7 classes. 2.) The minimum data entry is 59 and the maximum data entry is 450, so the range is = 391. Divide the range by the number of classes and round up to find the class width. 391/7 = when rounded up, round to 56. The reason we do this is because we want to know how to evenly divide up the numbers from evenly into 7 groups. Range 391 ; Class Width ) Use the minimum data entry as the lower limit for the first class. To find the lower limits of the remaining six classes, add the class width of 56 to the lower limit of each previous class. 4.) Then tally the numbers that go into each interval. 5.) Add up your tallies and write the frequency as a number in your table. Classes Tallies Frequency

5 Try it Yourself 1, Pg. 40 Construct a frequency distribution using the ages of the 50 richest people data set listed below. Use eight class. Solution: 1.) 8 classes 2.) Range: = 54 Class Width: 54/8 = ) 4.) and 5.) Classes Tallies Frequency

6 The above frequency distributions are called standard frequency distributions. You can also include other features such as midpoint, relative frequency, and cumulative frequency. Midpoint is the sum of the lower and upper limits of the class divided by two. Also called the class mark. Formula: lower limit+upper limit 2 Relative Frequency is the portion or percent of the data that falls in that particular class. To find the relative frequency, divide the frequency f, by the sample size n. Formula: frequency total number of data entries = f n Cumulative Frequency is the sum of the frequencies of that class and all previous classes. The cumulative frequency of the last class is equal to the sample size n. You only have to find class the first midpoint using the formula. Then you can just add the class width to the previous midpoint to find the next midpoint. Examples Pg.41, Example 2 and Try it Yourself 2

7 Example 2: Using the frequency distributions you found in example 1, find the midpoint, relative frequency, and cumulative frequency of each class. Identify any patterns. Solution: Classes f Midpoint Relative f Cumulative f (59+114)/2 5 = =.166 = 17% ( )/2 8 = =.266 = 27% 5+8 = ( )/2 6 = =.2 = 20% = ( )/2 5 = =.166 = 17% = ( )/2 2 = =.066 = 7% = ( )/2 1 = =.033 = 3% = ( )/2 3 = =.1 = 10% = 30

8 Try it yourself 2 Using the frequency distribution from try it yourself 1, find the midpoint, relative frequency, and cumulative frequency of each class. Identify any patterns. Solution: Classes f Midpoint Relative f Cumulative f (35+41)/2 = =.04 = 4% (42+48)/2 = =.1 = 10% 5+2 = (49+55)/2 = =.14 = 14% = (56+62)/2 = =.14 = 14% = (63+69)/2 = =.2 = 20% = (70+76)/2 = =.1 = 10% = (77+83)/2 = =.16 = 16% = (84+90)/2 = =.12 = 12% = 50

9 2.1 Part 2 Graphs of Frequency Distributions There are four different graphs to display frequency distributions. 1.) Frequency Histogram 2.) Frequency Polygon 3.) Relative Frequency Histogram 4.) Cumulative Frequency Ogive 1.) Frequency Histogram is a bar graph that represents the frequency distribution of a data set. It must have the following properties. The horizontal scale (across or the x-axis) is quantitative and measures the data values. The vertical scale (up and down or the y-axis) measures the frequencies of the classes. Consecutive bars (bars next to each other) much be touching. Because the bars of a histogram touch they must begin and end at class boundaries, which are the numbers that separate classes without gaps between them. If data entries are integers (,-3, -2, -1, 0, 1, 2, 3, ) subtract 0.5 from each lower limit to find the lower class boundary and add 0.5 to each upper limit to find the upper class boundary.

10 Steps to Creating a Histogram 1.) Draw a quarter plane, only positive x and positive y axis. 2.) Label x-axis with lower and upper limits of your frequency distribution or with the midpoints of each class. 3.) Label y-axis with numbers for the frequency. 4.) Make a bar for each interval that goes up to the frequency of that interval. *MAKE SURE ALL BARS TOUCH* Examples; Example 3 and Try it Yourself 3 on Pg Example 3: Draw a frequency histogram for the frequency distribution in Example 2. Describe any patterns. Frequency distribution is show below. Classes f

11 Pattern: Over half the GPS navigators are priced below $ Try it Yourself 3: Draw a frequency histogram for the frequency distribution in try it yourself 2. Describe any patterns. Frequency distribution is show below. Classes f Midpoint Relative f Cumulative f % % % % % % % % 50

12 Pattern: The most common age bracket for the 50 richest people is Frequency Polygon is a line graph that emphasizes the continuous change in frequencies. A frequency polygon is another way to graph a frequency distribution. Steps to Constructing a Frequency Polygon 1.) Draw a quarter plane, only positive x and positive y axis. 2.) Label x-axis with the midpoint of your frequency distribution and subtract the class width from the first class midpoint and add the class width to the last midpoint to extend the graph to the left and right so that the beginning and ending points touch the x-axis so that it creates a polygon. 3.) Label y-axis with numbers for the frequency. 4.) Place points that correspond to the given values and connect with lines (making a line graph).

13 Examples, Pg.43 Example 4 and Try it Yourself 4 Example 4 Classes f

14 Try it Yourself 4 Classes f Midpoint Relative f Cumulative f % % % % % % % % 50

15 There are two other types of graphs you can use to represent frequency distributions. Relative Frequency Histogram is a histogram which graphs the relative frequencies rather than the actual frequencies the y-axis is labeled with the decimal equivalent of the percentage, not the percentage itself. Steps to Constructing a Relative Frequency Histogram 1.) Draw a quarter plane, only positive x and positive y axis. 2.) Label x-axis with lower and upper limits of your frequency distribution or with the midpoints of each class. 3.) Label y-axis with the relative frequencies of each class (in decimal form). 4.) Make a bar for each interval that goes up to the frequency of that interval. *MAKE SURE ALL BARS TOUCH* Example

16 Ogive is a line graph which graphs the cumulative frequency of a frequency distribution. Should always go upwards. Steps to Constructing an Ogive (Cumulative Frequency Graph) 1.) Draw a quarter plane, only positive x and positive y axis. 2.) Label x-axis with lower and upper limits of your frequency distribution or with the midpoints of each class. 3.) Label y-axis using a scale that will contain all cumulative frequencies. 4.) Plot points that correspond to the data values and connect the points with lines (making a line graph).

17 2.2 More Graphs and Displays Part 1 Graphing Quantitative Data Sets Vocabulary Stem-and-leaf plot Stem Leaf Dot Plot exploratory data analysis (EDA) The ways we learned to display quantitative data in 2.1 are the traditional ways to display the data. Two of the newer ways we will learn in this section. o 1.) Stem-and-Leaf Plot o 2.) Dot Plot Stem-and-Leaf Plot is a two column table with the stems on the left side and the leaves on the right. EDA was created by John Tukey in Stem are all the digits to left of the last digit of a number. Leaf are the last digits of a number.

18 Steps to make a stem-and-leaf plot 1.) Determine the highest and lowest numbers in the data set and figure out what all digits are other than the digit on the right end. 2.) Put the digits, other than the last one, in order from least to greatest under the stem side of your plot. 3.) Place the last digits that match the first ones in order on the leaf side of your plot. Solution

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20 To construct a stem and leaf plot with two entries for each stem 0-4 will go with the first entry and 5-9 will go with the second. Solution

21 Dot plot is a number line where the data entries are plotted as dots over the correct number on the number line. To make a dot plot you must have a number line and use dots. A dot plot allows you to see how data are distributed, determine specific data entries, and identify unusual data values. Steps to Making a Dot Plot 1.) Draw a number line that will include numbers from your smallest number to your largest numbers, usually in increments of 1. 2.) Place a dot above each number the amount of times that that numbers appears in your data set. Solution

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23 2.2 More Graphs and Displays Part 2 Graphing Qualitative Vocabulary Pie Chart Pareto Chart Paired Data Sets Scatter Plots Time Series Time Series Chart We will learn 2 ways to graph qualitative data o 1.) Pie Chart o 2.) Pareto Chart Pie Charts A pie chart is a circle that is divided into sectors that represent categories and the percentages/frequencies of data in those categories. Steps to make a pie chart 1.) Begin by finding the relative frequency or percent of each category. 2.) Then use the percentage to determine the central angle that corresponds to each category. To find this you multiply the relative frequency by ) Use a compass to draw your circle and a protractor to mark off the angles. 4.) Label the pie chart.

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25 Solution: data is a Pareto Chart. Another way to graph qualitative Pareto Charts

26 A Pareto Chart is a vertical bar graph in which the bars represent the frequency or relative frequency of a category rather than an interval, The bars of a Pareto Chart are in decreasing order of height to help highlight important data. Steps to Creating a Pareto Chart 1.) Make a quarter plane. 2.) Label the vertical axis with frequency or relative frequency. 3.) Graph each bar so that they are NOT touching and at the height that matches the frequency of the category being graphed. 4.) Make sure to label each bar with the category that is represents.

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29 2.2 More Graphs and Displays Part 3 Graphing Paired Data Sets Vocabulary Paired Data Sets Scatter Plots Time Series Time Series Chart We will learn 2 ways to graph qualitative data o 1.) Scatter Plot o 2.) Time Series Chart Scatter Plot When each entry in one data set corresponds to one entry in a second data set the sets are called paired data sets. For example, one data set may contain the costs of an item and the second data set may contain how many units of the item were sold at each different cost. The two data sets corresponded/ are related. A scatter plot is a graph in the coordinate plane where the data sets are plotted as ordered pairs. How to Create a Scatter Plot On The Calculator 1.) Hit STAT and go to #1 Edit (or just hit enter because it is already on it). 2.) In L1 put in the first data set and in L2 put in the numbers from the 2 nd data set that correspond to each number in L1. 3.) Hit y= right under the screen and go up and press enter to highlight Plot 1. 4.) Now hit ZOOM and go down to Stat Plot. Now you should see your scatter plot on the screen.

30 Interpreting a Scatter plot 1.) Look to see what the vertical and horizontal axis of the scatter plot represent. 2.) Look for a correlation between the two. Does one increase as the other increases? One decrease as the other increases? One increase as the other decreases? 3.) Then write the correlation in words. Solution:

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32 Time Series and Time Series Chart A time series is a set of quantitative data (measures) taken are regular intervals over a period of time. For example the amount of rain each day for a month would be a time series. Each day being the time and the amount of rain being the quantitative measure. Graphing a time series you use a time series chart. A time series chart is a line graph in which the time is usually placed on the x-axis (or horizontal axis) and the quantitative data is usually placed on the y-axis (or vertical axis). Creating a Time Series Chart 1.) Determine the intervals for the horizontal axis (the times you are given whether it be years, minutes, hours, etc.) and for the vertical axis (the measures that you are given or have taken). 2.) Draw a coordinate plane, whether it be a quarter plane or full plane.

33 3.) Mark off the numbers on and label the horizontal and vertical axis. 4.) Now plot each point and connect them in order with line segments to create a line graph. 5.) Now you can look at the data to determine if the quantitative measure increases or decreases over time, if there are any trends, if it increases an decreases in certain intervals, etc. Solution:

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