with many modifications Statistics Statistics – thanks to www.k8accesscenter.org/documents/JKnight.webinar.ppt with many modifications

Statistics is the science of data. What is Statistics? Statistics is the science of data. As mathematicians, we help the scientists organize, simplify and present their data. What is data? Data are numbers with a context (a story).

THE SAME DATA CAN YIELD DIFFERING RESULTS Statistics THE SAME DATA CAN YIELD DIFFERING RESULTS Click on the man for an example.

So we can present the best statistics We need to know our data So we can present the best statistics

Puzzle – Are you really awake? Statistics Puzzle – Are you really awake? A certain five letter word becomes shorter when you add two letters to it. What is the word? Short SHORT

Measures of Central Tendancy Statistics Measures of Central Tendancy In some cases (not the airlines one) we can replace tables of data with one number. This is helpful to the reader – easier to understand. These are called “measures of central tendancy” Does our data tend toward a central number?

Means, Medians, Modes and Ranges Statistics Mean = Average of the data. -Add data elements and divide by the number of elements Median = Middle value of data -Arrange data in order, choose the middle value -If there is no middle number, the Median is the average of the middle numbers Mode = Value that occurs the most -Data with no repeats has no mode -Several repeats occurring same number of times has several modes Range = High value minus the low value Example 1-1a

Means, Medians, Modes and Ranges Statistics Means, Medians, Modes and Ranges What is the Mean, Median, Mode and Range for the data? 8, 7, 20, 8, 22, 7, 7, 9, 4, 28 Mean = 12 Which of these is the best representation of the data? Median = 4, 7, 7, 7, 8, 8, 9, 20, 22, 28 Since there are two middle numbers take the average of the two middle numbers which is 8, The median is 8 Mode = 7 Range = 28-4 = 24

Means, Medians, Modes and Ranges Statistics Means, Medians, Modes and Ranges For the first three sets of numbers below, the value of x represents the same type of measurement for central tendency. Find the value of x for the fourth set of numbers. 4 8 16 28 40 52 x=22 13 13 17 23 29 29 x =___ 5 15 20 25 50 x=20 16 22 22 22 24 x=22 20 What type of measurement does x represent?___________ Median

Central Tendency – Find data Statistics Central Tendency – Find data Catherine scored 152 and 165 for her first two bowling games. What score will she have to achieve on her third game to have an average of 170?

So, let’s study the pictures!!!! Statistics Sometimes we try to replace the data with a number. Sometimes it’s better to use a picture. So, let’s study the pictures!!!!

The picture will help us answer the question– where is the data? Statistics Statistical Pictures The picture will help us answer the question– where is the data?

Statistics Frequency Pictures There are a few pictures that tell us frequency (we will start with 2 and add one later) Frequency is used for large sets of data when we only have a few choices. We want to be able to easily see which is the most popular choice. We want to easily see the range, mode and median of the data (we can see all of these from Frequency Pictures). We also want to see if the data is clustered around a choice.

The first data display you probably ever used was a Statistics The first data display you probably ever used was a TALLY CHART This is also called a FREQUENCY TABLE. http://www.eduplace.com/math/mw/background/4/09/te_4_09_overview.html

What is the most popular snack? Read the tally chart. What is the most popular snack? What is the least popular snack? NUTS Jello or potato chips

Frequency distributions and line plots Statistics Frequency distributions and line plots Frequency distribution = a display of how FREQUENTLY something occurred. Line plots = a graphical display using a number line to show data. Ex1. The pulse rates of students before physical education class are recorded in the table. Pulse Rate 68 69 70 71 72 73 74 75 76 77 Number of Students 3 4 0 0 5 2 1 4 3 1 x x x x x x x x x x x x x x x x x x x x x x x 68 69 70 71 72 73 74 75 76 77 How many students had a pulse rate lower than 74? _______ 14

Frequency distributions and line plots Statistics Frequency distributions and line plots Ex3. The numbers below represent the ages of the players on a softball team. Make a line plot to show the data. 29, 26, 25, 32, 27, 25, 32, 26, 24, 25 xxx xx xx x x x 23 24 25 26 27 28 29 30 31 32 33 What numbers do we need on the number line? _______ 24-32 Line Plots – quick and easy way to see how data is distributed – just make an “x” above the number.

Collect data and put in chronological order Statistics Making a Line Plot Collect data and put in chronological order Ex. Scores on a math test 63 68 72 74 79 80 80 83 84 84 84 85 88 90 90 90 90 93 95 95 95 97 Determine a scale and intervals If you have a small range, you should probably use intervals of 1 With larger scales, it is best to mark intervals every 2, 5, or 10 numbers. Ex. Scores on a math test Use a scale from 60 to 100 and intervals of 5

Draw a horizontal line and mark the intervals Statistics Making a Line Plot Draw a horizontal line and mark the intervals Ex. Scores on a math test 63 68 72 74 79 80 80 83 84 84 84 85 88 90 90 90 90 93 95 95 95 97 60 65 70 75 80 85 90 95 100

Mark an X above the number for each data point Statistics Making a Line Plot Mark an X above the number for each data point If a number is repeated, place one X above the other Count the number of data points and count the number of X’s to make sure you plotted each point! Ex. Scores on a math test 63 68 72 74 79 80 80 83 84 84 84 85 88 90 90 90 90 93 95 95 95 97 x x x x x x x x x x x x x x x x x x x x x x 60 65 70 75 80 85 90 95 100

Understand the scale and interval Statistics Reading a Line Plot Understand the scale and interval Ex. This line plot has intervals of 2, so an X that falls between numbers would represent the median of those numbers. An X between the 6 and 8 would represent “7” X X X X X X X X X X X X X 0 2 4 6 8 10 12 14 16

Reading a Line Plot Finding the Range Statistics Reading a Line Plot Finding the Range Range – difference between highest value and lowest value Subtract the number represented by the first X from the number represented by the last X Do NOT just look at the numbers on the scale. The Range is NOT 16 – 0 = 16 - = 14 The RANGE is 14 1 X X 15 X X X X X X X X X X X 0 2 4 6 8 10 12 14 16

Reading a Line Plot Finding the Median Statistics Reading a Line Plot Finding the Median Median – middle number when data is arranged in chronological order Cross off the first and last X. Continue crossing off the first and last X’s until you reach the middle. The MEDIAN is 8 X X X X X X X X X X X X X 0 2 4 6 8 10 12 14 16

Reading a Line Plot Finding the Median Statistics Reading a Line Plot Finding the Median Median – middle number when data is arranged in chronological order If there are 2 numbers left in the middle, the median is the middle of those 2 numbers. The 2 middle numbers are 7 and 8, so the median is 7.5 X X X X X X X X X X X X 0 2 4 6 8 10 12 14 16

Reading a Line Plot Finding the Median Statistics Reading a Line Plot Finding the Median Median – middle number when data is arranged in chronological order If you get confused crossing off the X’s, list the numbers represented by the X’s in chronological order and cross them off. X X X X X X X X X X X X X 0 2 4 6 8 10 12 14 16 1 2 4 4 7 8 8 8 10 10 10 11 15

Statistics Both line plots (or frequency plots) and tally marks (or frequency tables) help us GATHER and ORGANIZE as well as report data. We have better methods of COMMUNICATING our data.

Line graphs Bar graphs Histograms There are many different ways to communicate our data. Stem and Leaf plots Box and whisker graphs Line graphs Bar graphs Histograms Pie Graphs

Statistics Example The cafeteria wanted to collect data on how much milk was sold in 1 week. The table below shows the results. We are going to take this data and display it in 3 different types of graphs.

Bar Graphs Line Graphs & Picto-Graphs Statistics Bar Graphs Line Graphs & Picto-Graphs Tables, charts and graphs are convenient ways to clearly show your data.

How to Set Up a graph to communicate Data Statistics How to Set Up a graph to communicate Data We need to ask . . . What did we do (or change)? This goes on the horizontal part (or x-axis). Time (days, weeks, years) always goes on x-axis. What did we observe? This goes on the vertical part (or y-axis). This is what we measure. We want to see our results on the y-axis.

Statistics Bar Graph A bar graph is used to show relationships between groups. The two items being compared do not need to affect each other. It's a fast way to show big differences. Notice how easy it is to read a bar graph.

Statistics Line Graph A line graph is used to show continuing data; how one thing is affected by another. It's clear to see how things are going by the rises and falls a line graph shows.

Statistics Bar Graph The same data displayed in 3 different types of graphs. Line Graph Pictograph

Statistics On what day did they sell the most chocolate milk? Wednesday

Statistics Thursday On what day did they have a drop in chocolate milk sales?

Choosing the Right Graph Statistics Use a bar graph if you are not looking for trends (or patterns) over time; and the items (or categories) are not parts of a whole. Use a line graph if you need to see how a quantity has changed over time.  Line graphs enable us to find trends (or patterns) over time.

Statistics Bar Graphs Bar Graph = A graphical display representing data in different categories or groups. The length of a rectangle or bar is used to represent the numerical amount. Labels

Bar Graphs

Histograms Statistics Histogram = histogram is a display of intervals in a graph. A histogram shows what portion of cases fall into each of several categories. The categories are intervals (like time periods or ranges of data). The categories (bars) must be adjacent – TOUCH EACH OTHER!!! Score Distribution for 3-point contest Histograms often show time periods.

Line Graphs and Bar Graphs and Histograms must include: Statistics Line Graphs and Bar Graphs and Histograms must include: Title Labeled X and Y axes X –axis is the independent variable Time goes on the x-axis (unless in science we are measuring the time it takes for something to happen) Y-axis is what we observe (dependent variable) Equal intervals are used on both axes. Make sure you use them on the y-axis Key

Double Bar Graph Statistics The purpose of a double bar graph is to compare two or more sets of data.

Line graphs are useful to show a trend in data. Statistics LINE GRAPHS mste.illinois.edu Line graphs are useful to show a trend in data.

Double Line Graph Statistics A double line graph is used to compare two groups of related data over time.

Statistics LINE GRAPHS They can also show that there is no trend.

Line Graphs or Bar Graphs? Statistics Line Graphs or Bar Graphs? This data is better shown in a bar graph. Note – there are 4 different data sets represented on the graph – they just use different colors.

Statistics Sometimes they use shapes instead of colors.

Pictograph Statistics Days of the Week Milk served Monday Tuesday Wednesday Thursday Friday Milk served = about 10 cartons of chocolate milk = about 10 cartons of vanilla milk = about 10 cartons of strawberry milk

Pictograph All pictographs have a title. Statistics Pictograph All pictographs have a title. Rows and columns shape the pictograph. Label each row and column. Use pictures to show the data. Each picture equals a certain amount of data. Pictographs need a key.

Statistics Stem and Leaf Plots Stem and Leaf Plot allows you to display a large quantity of data, in a smaller space and still see each value. In a stem-and leaf plot, the greatest common place value of the data is used to form stems. The numbers in the next greatest place-value position are then used to form the leaves.

Prepare a stem and leaf plot for the scores on a recent test: 57 60 88 85 79 70 65 98 97 59 58 65 62 77 77 75 73 69 82 81 Notice that the data (numerical facts) are numbers between 57-98. Create the stem by listing numbers from 5-9. Rearrange the leaf in numerical order from least to greatest Stem Leaf Stem Leaf 8 5 6 7 8 9 7 9 5 6 7 8 9 7 8 9 5 5 2 9 0 2 5 5 9 9 7 7 5 3 0 3 5 7 7 9 8 5 2 1 1 2 5 8 8 7 Key: 7 9 means 79 7 8 Match up the data to the stem-and-leaf. The last digit in 57 will match up with the stem 5. Then the last digit in 60 will match up with the stem 6. Then the last digit in 88 will match up with the stem 8 and this pattern will continue until all data have been recorded in the stem-and-leaf.

Make a Stem and Leaf Plot Statistics Make a Stem and Leaf Plot Age of United states Presidents at their First Inauguration: 57 61 57 57 58 57 61 54 68 51 49 64 50 48 65 52 56 46 54 49 50 47 55 54 42 51 56 55 51 54 51 60 62 43 55 56 61 52 69 64 (Through the 40th presidency) Stem Leaf Key: 5 7 means 57 4 5 6 2 3 6 7 8 9 9 0 0 1 1 1 1 2 2 4 4 4 4 5 5 5 6 6 6 7 7 7 7 8 0 1 1 1 2 4 4 5 8 9

It is easy to interpret or analyze information from the Stem-and-Leaf. Statistics It is easy to interpret or analyze information from the Stem-and-Leaf. How many presidents were 51 years old at their inauguration? What age is the youngest president to be inaugurated? What is the age of the oldest president to be inaugurated? How many presidents were 40-49 years old at their inauguration? 23 42 69 7 Stem Leaf: Age of United States Presidents at their First Inauguration (through the 40th Presidency) Rearrange the leaf in numerical order from least to greatest 4 5 6 2 3 6 7 8 9 9 0 0 1 1 1 1 2 2 4 4 4 4 5 5 5 6 6 6 7 7 7 7 8 Key: 5 7 means 57 0 1 1 1 2 4 4 5 8 9

Statistics Stem-and-leaf plots Original data set 22, 25, 25, 31, 31, 31, 32, 35, 40, 40, 43, 44, 46, 47, 53, 53, 55, 58, 59, 59, 62, 62, 63 Find the median, mode and the range. Median = 44 Mode = 31 Range = 41

Statistics Stem-and-leaf plots What data is represented by the stem-and-leaf plot below? 9 10 11 12 3 6 8 0 3 6 1 3 9 9|3 = 93 {93,96,98,110,113,116,121,123,129} Stem and Leaf plots help us both gather and use our data. However, they are a bit confusing to read.

Puzzle – Can you say this correctly? Statistics Puzzle – Can you say this correctly? What word in the English language does every Harvard graduate pronounce wrong? Clue: You pronounce it wrong as well!

Last, but not least, is the pie chart. Statistics PIE CHARTS Last, but not least, is the pie chart. This CANNOT be used in all situations. We can only use this when we have parts of a whole. We divide up the circle to represent fractions or percentages of the total.

A pie chart shows a fraction of the total amount. It represents a size relationship between the parts and the whole. This graph represents the results of a study done by the University of California on the number of words read or heard by a person during a year. It is useful to show how TV and computer are the most. But we would not use the picture to get information about movies and recorded music (we would look at the chart.

Sometimes is it not the best tool. Pie Charts Sometimes is it not the best tool. These represent the same data. Which is easier to read?

Statistics PIE CHARTS http://www.fedstats.gov/kids/mapstats/sconcepts_pie.html For this class, about what percent is determined by tests? (HW = homework)

Using the pie chart, if we want to conserve water, Statistics Pie Charts Using the pie chart, if we want to conserve water, what should we try to modify first?

How to construct a Pie Graph Statistics How to construct a Pie Graph How to construct a pie graph. 1. Add up all of the data to find the total amount. 2. Write each data element as a fraction of the total. 3. Using a proportion, convert those to fractions of 360º 4. Make sure all of your degrees add to 360 (adjust if necessary). 5. Using a protractor and compass make a circle graph. 6. Give the graph a title and label the parts. Use a key if necessary.

Statistics

Statistics

Statistics

Statistics – making data easy to use