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Statistics

Statistics. Year 9. Note 1 : Statistical Displays. Note 1 : Statistical Displays. Note 1 : Statistical Displays. 3. Oliver. IWB Ex 31.01 Pg 859. Sally and Mark (with 4 each). Note 1 : Statistical Displays. Note 1 : Statistical Displays. Note 2: Dot Plots.

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Statistics

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  1. Statistics Year 9

  2. Note 1: Statistical Displays

  3. Note 1: Statistical Displays

  4. Note 1: Statistical Displays 3 Oliver IWB Ex31.01 Pg 859 Sally and Mark (with 4 each)

  5. Note 1: Statistical Displays

  6. Note 1: Statistical Displays

  7. Note 2: Dot Plots A dot plot uses a marked scale Each time an item is counted it is marked by a dot

  8. Dot Plots - Symmetry A symmetric distribution can be divided at the centre so that each half is a mirror image of the other.

  9. Dot Plots - Skewness

  10. Dot Plots - Outliers A data point that diverges greatly from the overall pattern of data is called an outlier.

  11. Dot Plots e.g. This graph shows the number of passengers on a school mini bus for all the journeys in one week. IWB Ex31.02 Pg 863 19 How many journeys were made altogether? What was the most common number of passengers? 6

  12. Note 3: Pie Graphs

  13. Note 3: Pie Graphs Pie Graphs are used to show comparisons ‘Slices of the Pie’ are called sectors Skills required: working with percentages & angles e.g. 20 students in 9Ath come to school by the following means: 10 walk 5 Bus 3 Bike 2 Car Represent this information on a pie graph.

  14. Note 3: Pie Graphs e.g. 20 students in 9Ath come to school by the following means: 10 walk 5 Bus 3 Bike 2 Car = 10 × 18° = 180° = 90° = 5 × 18° = 54° = 3 × 18° = 36° = 2 × 18° All 20 Students represent all 360°of a pie graph How many degrees does each student represent? = 18°

  15. Note 3: Pie Graphs IWB Ex31.03 Pg 870 We can also use percentages and fractions to calculate the angles e.g. 500 students at JMC were surveyed regarding their TV provider at home. 180 had Skyview, 300 had Freeview and 20 had neither. Represent this in a pie chart. × 360° = 129.6° × 360° = 216° × 360° = 14.4°

  16. Note 4: Stem & Leaf Graphs Daily absences from JMC for a six week period in Term 3 are as follows:

  17. Note 4: Stem & Leaf Graphs Daily absences from JMC for a six week period in Term 3 are as follows: These figures can be summarized in a stem and leaf graph

  18. Note 4: Stem & Leaf Graphs

  19. Note 4: Stem & Leaf Graphs IWB Ex31.04 Pg 875

  20. Note 5: Scatter Plot IWB Ex31.05 Pg 879 Eg: this has a positive relationship – the taller the person the longer they can jump Scatter Plots show the relationship between two sets of data.

  21. Note 6: Time Series Graph This ‘line graph’ shows what happens to data as time changes Time is always on the x-axis Data values are read from the y axis What are some of the features of this graph? # of advertisements Time

  22. Note 6: Time Series Graph Each week, roughly the same amount of advertisements are sold The most popular days to advertise are: Wednesday & Saturday The least popular days to advertise are: Monday & Tuesday What are some of the features of this graph? IWB Ex31.06 Pg 884 IWB Ex31.05 Pg 879

  23. Calculating Statistics - averages • Mean (average) – The mean can be affected by extreme values • Median – middle number, when all data is placed in order. Not affected by extreme values • Mode – the most common value/s

  24. Note 7: Mean • Mean (average) – The mean can be affected by extreme values x =

  25. Note 7: Median • Median – middle number, when all the number are placed in order. Not affected by extreme values

  26. Note 7: Median • Median – middle number, when all the number are placed in order. Not affected by extreme values

  27. Note 7: Mode • Mode – is the most common value, one that occurs most frequently e.g. Find the mode of the following

  28. Note 7: Calculating Averages • In statistics, there are 3 types of averages: • mean • median • mode Mode Median Mean - x The middle value when all values are placed in order The most common value(s) Affected by extreme values Not Affected by extreme values IWB Ex31.07 Pg 892 Ex31.08 Pg 896 Ex31.09 Pg 901

  29. Starter • Calculate the mean for each of the following: a) 4, 8, 12, 4, 1, 1 b) 40, 50 c) 21, 0, 19, 20 • Ten numbers add up to 89, what is their mean • Calculate the mean to 2dp a) 84, 31, 101, 6, 47, 89, 49, 55, 111, 39, 98 b) 1083, 417, 37.8, 946 • A rowing ‘eight’ has a mean weight of 86.375kg. Calculate their combined weight • A rugby pack of 8 schoolboy players with a mean weight of 62kg is pushing against a pack of 6 adult players with a mean weight of 81kg. Which pack is heavier? Explain why?

  30. Note 8: Frequency Tables A frequency table shows how much there are of each item. It saves us having to list each one individually. 8 2 4 56, # of houses

  31. Note 8: Frequency Tables How would you display this information in a graph?

  32. Note 8: Frequency Tables Tables are efficient in organising large amounts of data. If data is counted, you can enter directly into the table using tally marks e.g 33 students in 10JI were asked how many times they bought lunch at the canteen. Below is the tally of individual results. 0 4 0 3 5 0 5 5 0 2 1 0 5 2 3 0 0 5 5 1 2 5 5 3 0 0 1 5 0 5 1 3 0 The data can be summarised in a frequency table

  33. Note 8: Frequency Tables IWB Ex31.11 Pg 910 Calculate the mean = = = = 2.3 Why is this mean misleading? Most students either do not buy their lunch at the canteen or buy it there every day. Total 33

  34. Starter • Write down the median of each of these sets of numbers a) {12, 19, 22, 28, 31} b) {0, 6, 9, 11, 19, 20} • Write down the mode foe each of these sets of numbers a) {6, 8, 9, 9, 10, 6, 7, 9, 8} b) {4, 6, 8, 6, 4, 8} c) {3, 1, 0, 1, 5, 0, 6} • A roadside stall has some avocados for sale at $2 a bag. These are the coins in the ‘honesty’ box on Tuesday. 5 x 20c coins 2 x 50c coins 2 x $1 coins 1 x $2 coins a) what is the median of the coins b) on Wednesday there were 24 coins in the box. The mean value of the coins was 25cents. Which gives better information about the number of bags sold – the mean or the median. 22 10 9 No mode Two modes are 0 and 1 35 cents The mean gives information about the total sold: 24 x 25cents = $6. 3 bags were sold

  35. Note 9: Histograms When a frequency diagram has grouped data we use a histogram to display it - measured data (e.g. Height, weight)

  36. Note 9: Histograms When a frequency diagram has grouped data we use a histogram to display it

  37. Note 9: Histograms IWB Ex31.12 Pg 916 When a frequency diagram has grouped data we use a histogram to display it

  38. Calculating Statistics • Range – a measure of how spread out the data is. The difference between the highest and lowest values. • Lower Quartile (LQ) – halfway between the lowest value and the median • Upper Quartile (UQ) – halfway between the highest value and the median • Interquartile Range (IQR) – the difference between the LQ and the UQ. This is a measure of the spread of the middle 50% of the data.

  39. Example: e.g. 40, 41, 42, 43, 44, 45, 49, 52, 52, 53 UQ median LQ Range = Maximum – Minimum = 53 – 40 = 13 IQR (Interquartile Range) = UQ – LQ = 52 – 42 = 10

  40. The five key summary statistics are used to draw the plot. Note 10: Box and Whisker Plots Minimum LQ Median UQ Maximum

  41. Note 10: Box & Whisker Plot Comparing data Male Female x median minimum maximum Upper quartile extreme value Lower quartile IQR

  42. e.g. 1 1 2 2 3 3 4 4 4 5 6 7 18 LQ UQ The following data represents the number of flying geese sighted on each day of a 13-day tour of England 5 1 2 6 3 3 18 4 4 1 7 2 4 Find: a.) the min and max number of geese sighted b.) the median c.) the mean d.) the upper and lower quartiles e.) the IQR f.) extreme values Min – 1 Max - 18 Order the data - 4 Add all the numbers and divide by 13 – 4.62 (2 dp) LQ – 2 + 2 = 2 UQ – 5 + 6 = 5.5 2 2 5.5 – 2 = 3.5 18

  43. Note 11: Quartiles e.g. Calculate the median, and lower and upper quartiles for this set of numbers 35 95 29 95 49 82 78 48 14 92 1 82 43 89 Arrange the numbers in order 1 14 29 35 43 48 49 78 82 82 89 92 95 95 LQ UQ median Median – halfway between 49 and 78, i.e. = 63.5 LQ – bottom half has a median of 35 UQ – top half has a median of 89

  44. Starter

  45. Starter

  46. Summary: Data Display Line Graphs – identify patterns & trends over time Interpolation - Reading in between tabulated values Extrapolation - Estimating values outside of the range Looking at patterns and trends 0 1 2 3 4 5 6 7 8 9 10 11

  47. Summary: Data Display Pie Graph – show proportion Multiply each percentage of the pie by 360° 60% - 0.6 × 360° = 216° Scatter Graph – show relationship between 2 sets of data Plot a number of coordinates for the 2 variables Draw a line of best fit - trend Reveal possible outliers (extreme values)

  48. Summary: Data Display Histogram– display grouped continuous data – area represents the frequency frequency Bar Graphs– display discrete data Distance (cm) – counted data – draw bars (lines) with the same width – height is important factor

  49. Summary: Data Display Stem & Leaf – Similar to a bar graph but it has the individual numerical data values as part of the display – the data is ordered, this makes it easy to locate median, UQ, LQ 3 3 4 8 5 10 9 8 8 3 11 2 3 6 7 8 Back to Back Stem & Leaf – useful to compare spread & shape of two data sets 4 2 0 12 1 9 9 3 3 13 0 2 2 14 5 Key: 10 3 means 10.3

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