Statistics

1. Statistics Achievement Standard 1.10

2. The Statistical Enquiry Cycle PROBLEM This is where you decide what you would like more information on. • CONCLUSION • What did you learn about your investigation • What do the graphs say • What differences are there in the statistics • Can you infer that the difference in the sample is also in the population • Are there new problems to investigate • PLAN • You need to know what you will measure and how you will do it. • What data do you need • How will you collect it • What will you record • How will you record it • ANALYSIS • This is where you look at the data to see what it tells you about your problem. • Graph data and collect statistics DATA This is where your data is collected, managed and organised.

3. The Statistical Problem What is it that you would like information on, considering information is often more interesting when one group is different to another. Remember that when comparing groups we should look to use numeric data or qualitative groups. Once you have decided what you would like information on you need to write a question fully describing what you are interested in investigating. There are two ways in which you can write a question. Two example questions investigating bag weights could be: e.g. ‘I wonder if there is a difference between the bag weights of Year 11 boys and Year 11 girls at CHS’ e.g. ‘’Do Year 11 boys at CHS tend to have heavier bag weights than Year 11 girls at CHS’

4. The Plan Now that you know what you are going to investigate, you will need to find variables that will illustrate the differences. Sometimes there are restrictions on what data can be collected or the ease with which data can be collected. You may need to consider how you will collect and record the data. Data is often best recorded in a table. There are web sites from which you can extract data and a good starting point is asking yourself what differences you will expect. Remember: This Achievement Standard is an investigation into multivariate data which implies that you will initially look at a number of variables before deciding which you will look for differences between

5. The Data It is important that the largest sample that is possible is used. Recommended minimum is 30 as drawing conclusions from small samples is suspect. Census At School is an excellent resource that contains a huge amount of interesting statistical information. It is likely this data will be used. When you collect data from website, you may need to ‘clean’ it. This refers to the process of removing invalid data points from your sample. e.g. When using data regarding bag weights, if an individual has no bag, do not record it as a 0. e.g. Check units are consistent, all 0 entries and always be suspicious of any data that seems out of place.

6. MEASURES OF CENTRAL TENDENCY The Analysis – Statistics and Graphs 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

7. 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

8. 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

9. MEASURES OF SPREAD 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.

10. Standard Deviation – is the measure of the average spread of the numbers from the mean. – for Year 11, your only concern is that the bigger the value, the more spread the data is. Quartiles – 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 and describes the middle 50% ¼

11. 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

12. 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

13. Stem and Leaf Plots – records and organises data – most significant figures form the stem and the final digits the leaves – 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

14. Calculating Statistics from Stem and Leaf Plots 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 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!

15. Note: Use the minimum and maximum values to determine length of scale Box and Whisker Plots – shows the minimum, maximum, LQ, median and UQ – ideal for comparing two sets of data 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?

16. Histograms – display grouped data – frequency is along vertical axis, group intervals are along horizontal axis – 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.

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

18. Use the data to determine scale to use on both axes Scatter Plots – looks for a relationship between two measured variables Outliers can generally be ignored – points are plotted like co-ordinates 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?

19. Time Series – 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 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

20. The Conclusion (and Making an Inference) When you have finished your analysis, it is important to see if you can make an inference. An inference is when you make a generalised statement about the whole population (generally if there is a difference), by using the findings from your sample. It is easiest if you write your statement in the following manner: “From my sample data I can make an inference about the population that… Year 11 Boys tend to be taller than Year 11 Girls (example). This is because… You will need to justify this inference by using your findings from both your statistics and/or graphs. Ways to justify your inference will be shown on an extra handout. Once you have made your inference with justifications, you should end off your investigation by making a conclusion. Your conclusion MUST answer your original question and can be written in the format: “Therefore I can conclude that… Year 11 boys at CHS are typically taller than Year 11 girls at CHS (example).

21. 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

22. Limitations and Improvements 1. In terms of Data Collection Typical Limitations Improvements - Sample too small - Obtain a bigger sample - Not random or representative - Get a representative sample - Taken over too short a time period - Take data over a longer time period - Outliers distort data - Ignore extreme outliers 2. In terms of Your Process Typical Limitations Improvements • Not enough statistics • calculated • Calculate more statistics • Not enough graphs used, • data could be compared better • Use other graphs (i.e. comparative histograms) • Scales on graphs too large • Change scales on graph (smaller) • Way graphs are drawn • Alter the way the graphs may be drawn