Statistics

1. Statistics

2. Measures of Central Tendency (Averages) Sum of all values 1. Mean - easy to calculate but is affected by extreme values - to calculate use: Total number of values Push equals on calculator BEFORE dividing e.g. Calculate the mean of 6, 11, 3, 14, 8 6 + 11 + 3 + 14 + 8 42 Mean = = = 8.4 5 5 e.g. Calculate the mean of 6, 11, 3, 14, 8, 100 6 + 11 + 3 + 14 + 8 + 100 142 Mean = = = 23.7 (1 d.p.) 6 6

3. a) for an odd number of values, median is the middle value e.g. Find the median of 39, 44, 38, 37, 42, 40, 42, 39, 32 32, 37, 38, 39, 39, 40, 42, 42, 44 To find placement of median use: n + 1 2 n = amount of data 2. Median - middle number when all are PLACED IN ORDER (two ways) - harder to calculate but is not affected by extreme values 9 + 1 = 10 = 5 2 2 Cross of data, one at a time from each end until you reach the middle value. OR Median = 39 b) for an even number of values, median is average of the two middle values e.g. Find the median of 69, 71, 68, 85, 73, 73, 64, 75 64, 68, 69, 71, 73, 73, 75, 85 n + 1 = 8 + 1 = 4.5 2 2 Median = 71 + 73 = 144 = 72 2 2 OR

4. 3. Mode - only useful to find most popular item - is the most common value (can be none, one or more) e.g. Find the mode of 188, 93, 4, 93, 15, 0, 100 15 Mode = 15 and 93 Range - can show how spread out the data is - is the difference between the largest and smallest values e.g. Find the range of 4, 2, 6, 9, 8 lowest value highest value Range = 9 – 2 = 7 (2 – 9) Note: Its a good idea to write in brackets the values that make up the range.

5. e.g. Here are the number of fundraising tickets sold by 25 members of a Hockey team. Place data on a frequency table. Ungrouped Frequency Tables 3, 5, 0, 1, 0, 2, 5, 2, 4, 0, 1, 2, 3, 5, 7, 2, 3, 3, 1, 4, 3, 3, 2, 0, 1 - Useful when dealing with large amounts of discrete data IIII 4 0 x 4 = 0 IIII 4 1 x 4 = 4 IIII 5 2 x 5 = 10 IIII I 6 3 x 6 = 18 II 2 4 x 2 = 8 III 3 5 x 3 = 15 0 0 I 1 7 x 1 = 7 25 62 To find the mean, we need the sum of the ticket numbers multiplied by their frequencies, and divide this by the total frequency. Check total frequency matches question! Mean = sum of x.f . total frequency = 62 25 = 2.48 tickets

6. IIII 4 4 0 x 4 = 0 IIII 4 8 1 x 4 = 4 IIII 5 13 2 x 5 = 10 IIII I 6 3 x 6 = 18 II 2 4 x 2 = 8 III 3 5 x 3 = 15 0 0 I 1 7 x 1 = 7 25 62 To find the median, determine its position by using the previous formula. Now, by adding down the frequency column, locate position of median n + 1 = 25 + 1 = 13 2 2 Therefore: Median = 2 tickets To find the mode, look for the highest frequency Therefore: Mode = 3 tickets

7. 1. Discrete Data – usually found by counting, usually whole numbers Types of Data e.g. Number of cars passing the school 2. Continuous Data – usually found by measuring e.g. Weights and heights of students Data Display 1. Bar Graph – shows discrete data – must have GAPS between bars e.g. Beside are the number of times 28 students went out for dinner last month. Place data on a bar graph.

8. Don’t forget a title Note gaps between bars Or axis labels

9. 2. Dot Plots – are like a bar graph – each dot represents one item e.g. Plot these 15 golf scores on a dot plot 70, 72, 68, 74, 74, 78, 77, 70, 72, 72, 76, 72, 76, 75, 78 Range plot between lowest and highest values

10. 3. Pictograms – uses symbols to represent fixed numbers – key shows the value of the symbol e.g. Using an appropriate symbol, draw a pictogram displaying the number of hours per week spent completing homework for the following subjects. Hours of Study in a Week Science English KEY Maths 1 hour

11. 4. Pie Graphs – show comparisons – slices are called sectors – uses percentages and angles (protractor and compass) e.g. Students of a class arrived to school in the following manner. Show on a Pie Graph Walked = 6 Cycled = 5 Car = 4 Bus = 9 90° 75° 60° 135° To calculate angle of sectors use: Amount of sector x 360 Total Data Student Mode of Transport Car Walked Walked = 6 x 360 = 90 24 Cycled Note: Instead of labels, a key could also be used. Bus

12. 5. Strip Graph – shows the proportion of each part to the whole – should have a scale – linked to pie graphs Quartiles e.g. Using Pie Graph example, Strip Graph drawn could use a scale of1 cm = 2 students – are measures of spread which with the median splits the data into quarters – method used is similar as to when finding median When the data is in order: – the lower quartile (LQ) has 25% or ¼ of the data below it. – the upper quartile (UQ) has 75% or ¾ of the data below it. – the Interquartile Range (IQR) = UQ – LQ

13. e.g. Find the LQ, UQ and the interquartile range of the following data 6, 6, 6, 7, 8, 9, 10, 10, 11, 14, 16, 16, 17, 19, 20, 20, 24, 24, 25, 29 Note: always find the median first 20 + 1 = 21 = 10.5 2 2 or cross off data 10 + 1 = 11 = 5.5 2 2 LQ =8 + 9 = 17 = 8.5 2 2 OR UQ = 20 + 20 = 40 = 20 2 2 OR IQR = UQ - LQ IQR = 20 – 8.5 = 11.5 e.g. Find the LQ, UQ and the interquartile range of the following data 5, 6, 8, 10, 11, 11, 12, 15, 18, 22, 23, 28, 30 Remember, always find the median first 13 + 1 = 14 = 7 2 2 or cross off data As the median is an actual piece of data, it is ignored when finding the LQ and UQ 6 + 1 = 7 = 3.5 2 2 LQ = 8 + 10 = 18 = 9 2 2 UQ = 22 + 23 = 45 = 22.5 2 2 IQR = 22.5 – 9 = 13.5

14. – records and organises data – most significant figures form the stem and the final digits the leaves Stem and Leaf Graphs – can be in back to back form in order to compare two sets of data e.g. Place the following heights (in m) onto a back to back stem and leaf plot BOYS = 1. 59, 1.69, 1.47, 1.43, 1.82, 1.70, 1.73, 1.35, 1.76, 1.68, 1.62, 1.84, 1.45, 1.50, 1.54, 1.73, 1.84, 1.71, 1.66 GIRLS = 1. 44, 1.46, 1.63, 1.29, 1.48, 1.57, 1.51, 1.42, 1.34, 1.45, 1.57, 1.59, 1.42 Look at the highest and lowest data values to decide the range of the stem Unordered Graph of Heights Ordered Graph of Heights Boys Girls Boys Girls   1.8 1.8 1.7 1.7 1.6 1.6 1.5 1.5 1.4 1.4 1.3 1.3 1.2 1.2 4 ,4 ,2 4, 4, 2 1 ,3 ,6 ,3 ,0 6, 3, 3, 1, 0 6 ,2 ,8 ,9 3 9, 8, 6, 2 3 4 ,0 ,9 7, 1, 7, 9 9, 4, 0 1, 7, 7, 9 5 ,3 ,7 4, 6, 8, 2, 5, 2 7, 5, 3 2, 2, 4, 5, 6, 8 5 4 5 4 9 9 Place the final digits of the data on the graph on the correct side

15. For each statistic, make sure to write down the whole number, not just the ‘leaf’! Graph of Heights Boys Girls   1.8 1.7 1.6 1.5 1.4 1.3 1.2 Calculating Statistics from Stem and Leaf Graphs 4, 4, 2 6, 3, 3, 1, 0 9, 8, 6, 2 3 9, 4, 0 1, 7, 7, 9 When finding median, LQ and UQ, make sure you count/cross in the right direction! 7, 5, 3 2, 2, 4, 5, 6, 8 5 4 9 e.g. From the ordered plot state the minimum, maximum, LQ, median, UQ, IQR and range statistics for each side BOYS GIRLS Median = 13 + 1 = 7 2 Minimum: 1.35 m 1.29 m Maximum: 1.84 m 1.63 m LQ: 1.50 m 1.42 m LQ/UQ = 6 + 1 = 3.5 2 Median: 1.68 m 1.46 m UQ: 1.73 m 1.57 m IQR: 1.73 – 1.50 = 0.23 m 1.57 – 1.42 = 0.15 m Range: 1.84 – 1.35 = 0.49 m 1.63 – 1.29 = 0.34 m Remember: If you find it hard to calculate stats off graph, write out data in a line first!

16. Note: Use the minimum and maximum values to determine length of scale – shows the minimum, maximum, LQ, median and UQ – ideal for comparing two sets of data Box and Whisker Plots e.g. Using the height data from the Stem and Leaf diagrams, draw two box and whisker plots (Boys and Girls) Box and Whisker Plot of Boys and Girls Heights Males Minimum LQ Median UQ Maximum Females Question: What is the comparison between the boy and girl heights? ANSWER? EVIDENCE?

17. – used when dealing with a large amount of continuous data and groups are needed e.g. Listed below are the heights (in cm) of 25 students. Represent the data on a frequency table 167, 173, 171, 149, 162, 174, 185, 165, 160, 170, 173, 161, 158, 172, 168, 168, 178, 170, 180, 166, 183, 150, 164, 161, 164 Grouped Frequency Tables Note: Make sure you have enough groups but don’t make them too small! I 1 (140 + 149) / 2 144.5 144.5 x 1 144.5 II 2 154.5 309 IIII IIII I 11 164.5 1809.5 IIII III 8 174.5 1396 III 3 184.5 553.5 25 4212.5 To calculate the mean a midpoint is needed and the formula used is: e.g. Calculate the mean from the above data and state the modal interval Mean = 4212.5 25 = 168.5 cm Modal Interval = 160 – 169 cm

18. – display grouped data – frequency is along vertical axis, group intervals are along horizontal axis Histograms – there are NO gaps between bars e.g. Graph the grouped frequency table data about heights onto a histogram Note: The groups from the table form the intervals along the horizontal axis and the highest frequency determines the height of the vertical axis.

19. – Side by side histograms can also be used to compare data Question: What is the comparison between the female and male heights? ANSWER? EVIDENCE?

20. Use the data to determine scale to use on both axes – looks for a relationship between two measured variables Outliers can generally be ignored – points are plotted like co-ordinates Scatter Graph/Plot e.g. Below are the heights and weights of Year 7 boys. Place on a scatter plot. Line of best fit If points form a line (or close to) we can say there is a relationship between the two variables. ANSWER? What is the relationship between the boys height and weight? EVIDENCE?

21. – a collection of measurements recorded at specific intervals where the quantity changes with time. Features of Time Series a) Order is important with all measurements retained to examine trends b) Long term trends where measurements definitely tend to increase or decrease Time Series c) Seasonal trends resulting in up and down patterns What are the short and long term trends? ANSWER? EVIDENCE? e.g. Draw a time series graph for the following data: Join up each of the points

22. Good graphs should have: - an accurate heading (watch emotive headings) - scales in even steps Misleading Graphs - scales from zero unless a break is shown - values easy to read - bar graphs have the same width bars and similar shading Statistical Investigation Terms Population: The entire group of members under consideration Sample: When part of the group is surveyed Census: Whole population is surveyed Survey: Collection of information from some or all members of a population Sampling Frame: A list covering the target population A Good Sampling Frame: - should have each unit listed only once - has each unit distinguishable from others - is up to date

23. Investigations When planning an investigation: - think carefully about what you are trying to find (question) - what data is needed - how will you obtain the data - is the method practical and convenient - how will you record the information - how will you present the data

24. Choosing a Sample A sample should: 1) Be large enough to be representative of whole population 2) Have people/items in it that are representative of the population It is best to choose samples that are large and random but size may be affected by time, money, personnel, equipment etc. Some Sampling Methods Simple random sampling: 1- obtain a population list 2- number each member 3- use random table or random number on calculator Systematic sampling: 1- obtain a population list 2- randomly select a starting point on the list 3- select every nth member until desired sample size is reached Note: every nth member is found by: Population/group size Size of sample needed