MA30S APPLIED UNIT F: DATA MANAGEMENT CLASS NOTES

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1 1 MA30S APPLIED UNIT F: DATA MANAGEMENT CLASS NOTES 1. We represent mathematical information in more ways than just using equations! Often a simple graph or chart or picture can represent a lot of information. So how do we represent large amounts of mathematical information and data? FREQUENCY TABLES 2. Frequency tables just record how often different data values or categories occur. It is usually easiest to make a stroke tally as you count values and then just count the tally for each value. Eg: a frequency table from a survey of shoe sizes at our school: Shoe Size Tally Count Total 39 MA30S_F_DataMgmtClassNotes.doc Revised:

2 2 BAR CHARTS 3. Bar charts are useful also, they show frequencies or percentage of qualitative data (qualitative data is data that cannot be given a number such as a political party s name). Percentage of Popular Vote Votes by Party - Election Source: National Post 7 0 Conservative Liberal NDP Bloque Green Party 4. Recall that any chart should have a clear Title to explain what it is about. Also, the horizontal and vertical axis should be clearly labelled as to what they represent. And obviously, the source of the data should be included otherwise how do you know how reliable the information is? 5. Sometimes a bar graph is shown with horizontal bars: Percentage of Popular Votes Green Bloque NDP Liberal Conservative 7 Popular Vote Election Party Incidentally is very easy to make nice charts in any Spreadsheet program such as EXCEL.

3 3 PIE CHARTS Popular Vote Election Conservative Liberal 38 NDP 18 Bloque Green Pie charts show what percentage of the whole pie that each category is worth. What percentage of the vote did the Liberal party get? WHISKER AND BOX PLOTS 7. Most students will be familiar with these from their or their children s report cards. It is a convenient way to show the statistics of scores and how you or your child is doing. 20% Student s 40% 60% Mark 80% 100% Class Average 8. In a Whisker and Box plot it is rather evident how lots of data is arranged (like some class marks shown above). In the above case, marks ranged from 20% to 100%. The bottom quarter of the class was between 20% and 40%. The middle ranked half of the class was between 40% and 80%. The top quarter of the class was 80% to 100%, and the class average was about 70%. Notice the median (the middle ranked student) is 60%. (recall that median and average are seldom exactly the same thing!). And of course your or your child s mark is shown as about 75%, almost in the top quarter of the class!

4 4 9. Here is an important bar chart of all of Canada s population by age group from the Census in Can you notice some trends??? 1,600,000 1,400,000 Canadian Population by Age in 2002 Mean age of all sexes is Mean Age Male = (+) Mean Age Female = (+) Number 1,200,000 1,000, ,000 Male Female 600, , , Age Groups andover HISTOGRAMS 10. Histograms are just a picture of numerical data. The human eye usually sees patterns better in such a graph or picture. 11. A histogram of the above shoe size frequency table would look like this Shoe Sizes Histogram 12 Shoe Size Tally Count 7 4 Count Shoe Size Total 39

5 5 12. Notice that in a histogram that the area of each bar represents a percentage of the entire sample. Each category is the same width (one shoe size in this case). In the above case there was 39 samples. So the category for shoe size 9 takes up 10 out of 39 parts of the graph, or 25.64% of the data. In a histogram, there should not normally be any gaps between the bars in the categories (unlike a bar graph ). DISCRETE DATA COMPARED TO CONTINUOUS DATA 13. We sometimes distinguish between DISCRETE DATA and CONTINUOUS DATA. 14. Discrete data is represented by either whole numbers or distinct category values. Counting a number of books or labelling the months with numbers (Jan = 1, Feb = 2, etc) makes for Discrete Data. Number of Children per family Cyriac Roach Descendants or the number of children in families is discrete data. You can t have one and a half children! 15. If Teacher Bud Bundy had asked the students to count all the books on the shelves, the data would all have been whole number answers. These answers would be an example of discrete data, which are data you get when counting a finite number of distinct objects. (For example, there may be 71 or 72 books, but not 71.6 books.). Another example of discrete data: Counting all the days in a year that exceeded 30 C. 16. Continuous Data. Continuous data can have infinitely many precise values ( , , etc). Such data might include measurements of speed, area, time, or temperature. 17. An example of continuous data would be the value read by a precise thermometer. The data can have many fine values. Such data is called continuous data. Another example is measurement of time: there are intermediates values of time between 08:12 and 08:13, time does not just tick away by the minute with nothing in between. 18. In general, continuous data points can be connected by a line or a curve and we can interpolate intermediate readings. Discrete data points cannot be connected by a curve, since discrete values have no logical interpretation of intermediate values between data.

6 6 120 My Speed Driving to Kenora 100 Speed Elapsed Time [minutes] from when I left Home 19. Above is another example of continuous data. Every one-billionth of a second my GPS reported its speed on the highway to Kenora. 20. It would not be appropriate to connect the number of children my Acadian cousins had with a line since there is not such thing as a fraction of a child Family Size for Descendants of Cyriac Frequency Number of Children Advanced thinking. For purists, no instrument can actually be infinitely precise and every measure of continuous data in the real world involves reading an instrument. The instrument can only really give you a very fine category in which to put a reading. So in reality there is no such thing as continuous data that is derived from measurements. For example: even if your watch says that something happened at 08hours:12minutes:13.2 seconds, your watch only has limited precision, so how do you know the event didn t really happen at 08hours:12minutes: seconds? Or does nothing happen in between tenths of seconds in our universe?

7 7 SCATTER PLOTS 22. Scatter plots are another way to represent numerical data; between two sets of numbers. For example: years of school vs annual income. Thousands Annual Income Annual Income Vs Education Source: Survey of thirty people (Notional) Income = $6790.4* Years - $ Years of Education 23. Each dot (data point) here represents one person, it might be Bob, or June or Kyle. It might be you! Who had 15 years of school and made about $40K per year? dot) (Circle the Who had about 18 years of school and makes about $80K per year? (Circle the dot) 24. As you had seen in Grade 9 and 10 it is possible to find a correlation in numerical data. The trend shown above is that the longer you spend in school, the more your annual income. And you can even make close average formulas by regression to show that one s average income is on average: $7000 times years of school subtract $42,000. Of course if we survey a different 30 people the data may be slightly different. MAPS AND CONTOURS (ISOCLINES) 25. Geographic maps are a really great way to represent geographically related numbers. On a map for example one is often interested in the elevation, the height above sea-level, so they can sense where hills are and which way rivers flow!

8 8 26. In the map above (from around Swan River, Manitoba) the numbers represent all the points on the earth that are the same the elevation in feet above sea level; these continuous lines are called contours to a map maker or more correctly isoclines (points of equal elevation) to a mathematician. These lines on this map are connected at elevation intervals of every 25 feet. a. Mark on this map with the letter A the highest point. b. Mark on this map with the letter B the Highest point between Zaste Lake and Kyle Lake. c. Mark arrow heads on the map showing the direction the two creeks flow into Zaste Lake. Can you see a rule for finding what direction a river flows?. d. If you were in the middle of Zaste Lake in a boat, mark an area with a C where you would see the land making a prominent saddle shape. e. If you are standing in your boat in the west end of Zaste Lake can you see your buddy fishing in Kyle Lake just 0.4 miles away (or about 2,000 feet or 600 meters away)? (pretend there are no trees to block your view!) f. All the water on this map empties in the same direction here, in which direction does the water flowgo? g. If you were fishing in Kyle Lake, how many hill peaks would you see? (You may need to think of trigonometry or similar triangles for this question)

9 9 27. Contours are pretty hard to picture if you have lived on the flat prairie all your life as there is no significant elevation relief in the geography! Below is a map of Jasper, Alberta. Elevations are in meters. a. Mark the highest point on this map with an A. What is its elevation (meters): About how many feet is this elevation? : (3.28 feet = 1 meter) b. outline all the areas above 2000 meters by shading them in. c. Approximately what elevation is the town of Jasper (upper right corner of map on Highway 16)? : d. what direction is the river on the right flowing? (the Athabasca River). e. If you are sitting in a hot tub on the roof of your hotel in Jasper can you picture (using trigonometry and some conversions) what angle you are looking up at the peak at 2440 meters to the southwest? (1 meter = 3.28 feet, 1 mile = 5280 ft).

10 10 WEATHER MAPS 28. You see weather maps every day in the newspaper and on the TV. But what do all the symbols mean? And especially the lines? 29. This sort of weather chart should look familiar:

11 The symbols in any chart are usually explained with the chart in a Legend. This weather chart shows the temperatures in Canada on 8 Dec The different shades show the different temperatures in a region. The edge of each shaded colour is all the same temperature, or contour. All the points on the edge of each shaded area are the same temperature. All the places that are the same temperature, when connected with a line, are called isotherms. (Similar to isoclines, but isoclines means same elevation, isotherms means same temperature). 31. Highs and Lows. Notice the Hs and Ls on the chart. These stand for High pressure areas and Low pressure areas. These are significant because air moves from high pressure to low pressure. The differences in pressure in earth s atmosphere from distant place to distant place are usually less than 3%. But the difference is enough to get the air moving to try to move towards the low pressure area. A pressure of 1013 is considered an average normal. (this means 1013 milli-bars if you have taken Grade 10 Science and the weather unit). 32. Wind Direction. This wind chart shows some important information. It is from the same day and time as the previous chart. It shows the direction and the speed of the wind. In grade 12 you will study things that have a magnitude and direction, they are called vectors. The arrows in the chart show which way the wind is blowing, each longer tick on the tail of the arrow counts how many tens of miles per hour, any half-length tick represents is another five miles per hour. a. How fast is the wind over the Pas (YQD)?. What direction is the wind at YQD? b. How fast is the wind over Brandon (YBR)?.. Direction?. 33. Notice also that meteorologists always say from which way the wind is coming, not which way it is going. So if a meteorologist says the wind is a North wind, it is actually blowing south. The wind at Fort McMurray, YMM (where the tar sands are) is a northerly wind at 10 mph.

12 You might wonder why if the last two charts are the same date and time why the wind is not going from High to Low, it seems to be going perpendicular to the direction it should. A quick physics lesson would explain that, but it is because the earth is rotating. 35. Below is another busy map. Notice the contours on this map. These contours are called isobars. They connect points of equal pressure. Where the contours are really close is windy and likely stormy. 36. Other Symbols. Other symbols on the above map include the symbols for warm and cold fronts. Low pressure areas, especially with a front, are generally bad weather. a. Draw the symbol for a cold front (advancing cold air): b. Draw the symbol for a warm front (retreating cold air) 37. Now you have a good idea of how we use charts and graphs to represent complex mathematical information.