Wynberg girls high-Jade Gibson-maths-data analysis statistics
- 1. Data Analysis Chapter 10
- 2. Types of Data Quantitative data is data recorded with numbers eg: learner’s weight or number of goals Qualitative data is data recorded in words eg: favourite colours
- 3. … types of data cont. Within these two types of data we can also look at … Discrete data – information collected by counting (1, 2, 3 … no halves/quarters etc) Continuous data – information collected by measurement (may have decimals and fractions) Do Ex 10.8 Q1 (Pg 232)
- 4. Data Interpretation Once data has been collected and sorted, it has to be interpreted and analysed Two types of interpretation: Pictorial methods: involve drawing graphs
- 5. Arithmetic methods: involve working out: Measures of central tendency – mean median and mode Measure of dispersion – range, percentiles, quartiles and the interquartile range
- 6. Displaying data (Pictorial methods) Histograms – no gaps (quanitative data) Bar Graphs – bars do not touch Compounded bar graphs Dual bar graph – data displayed next to each other Sectional bar graph – data displayed ‘on-top of one another’ Pie Charts Broken line graphs
- 7. Ex. 10.2 (3)
- 8. 10.3 (1)
- 9. 10.4 (2)
- 10. Misleading graphs Ways graphs/charts can be misleading: Using 3D in pictograms/bar-charts Using perspective/shape to exaggerate Reversing the direction of an axis (to make a decrease seem like an increase) Altering the scale of the y-axis (to make it look more or less steep) Leaving part of the axis out to exaggerate differences http://www.coolschool.ca/lor/AMA11/unit1/U01L02.htm#
- 11. Misleading statistics Stats are notorious for being made up or misleading E.g.: during a political debate in USA , a member of the opposition claimed that employment had gone up during the President’s term of office; yes it had … but only because the population had increased, the number of unemployed people had also increased.
- 12. “ 86 % of statistics are made up on the spot and the remaining 24% are flawed”
- 13. Measures of central tendency Mean, mode and median “Averages”
- 14. … Mean (x) is like the average: Mean = sum of values number of values Can be affected by outliers, so not a good measure of central tendency if outliers
- 15. … Median is the one in the middle when placed in numerical order (smallest to biggest) If there are outliers then median is a better measure of central tendency Mode/Modal value is the value that appears the most
- 16. Things which can help with measures of central tendency Frequency tables Simple tables Or for grouped data Stem and Leaf diagrams – these are especially helpful for data with more than ten items
- 17. 10.9 (3) 5 19 8, 6, 3 20 7, 5, 5, 4, 0 0, 0 18 9, 9, 4 6, 6, 8, 8 17 9, 6, 3 2, 4, 5, 5, 7 16 0 6, 8, 8, 8 15 5, 1, 0 14 9, 6 13 9 12 4 11 0, 8 10 Robert Jabu Heights of mealie plants (in cm)
- 18. 10.10 (1) 9 2 0 9 0 10 0, 2, 2, 7 8 0, 2, 7, 8, 9 7 4, 5, 9 6 0, 1, 2, 2, 2, 4, 4, 6, 9, 9 5 4, 5, 5, 5, 5, 7, 7 4 2, 6, 9 3 Leaves Stem
- 19. Grouped data When the data has many different measurements involved in it, the data is usually grouped in intervals (classes). Try to have between 8 and 14 classes. And start with a value below the minimum in the data. Tally : lines used to count up the frequency of scores Frequency is the number of times that score/value appears
- 20. Example of a ‘Grouped data table’ Midpoint is the midpoint of that interval; calculated as on the table above fX = frequency multiplied by midpoint 48 8 6 //// / 6-10 9 (1+5) ÷2= 3 3 /// 1-5 fX (Frequency x midpoint) Midpoint (X) Frequency (f) Tally Classes
- 21. Analysing the grouped data We can calculate: Actual mean (x) = sum of values number of values Estimated mean (X) = sum of ‘fX’ values number of values We can draw a graph using the data: eg: a histogram with ‘classes’ on the x-axis and ‘frequency’ on the y-axis
- 22. … We can find both a mode and modal class: Mode: value that appears most Modal class: class (interval) with highest frequency We can estimate the median from a histogram: By estimating the value at which the ‘area’ of the histogram is divided into two equal parts
- 23. Histograms and frequency polygons Histograms and frequency polygons are both ‘frequency graphs’ The difference between them is that the histogram is made up of bars, whereas the frequency polygon is a line graph The ‘polygon’ is made from the lines of the graph and the horizontal axis
- 24. Drawing Frequency Polygons (2 methods) 1) Using the bars of a histogram Mark the midpoint of the top of each bar Join the points; including two points at zero on either side of the histogram
- 25. … 2) Without using a histogram: Plot the midpoint of each interval against the frequency Join the points; and add the two “zero” points on either side as with the histogram
- 26. Measures of Dispersion Tell us how the data is grouped around the “average” Is it closely grouped, or scattered widely? Measure of spread, scattering or dispersion of scores
- 27. Range Range = largest value – smallest value Has a few limitations in that it cannot be used for ‘grouped data’; and it doesn’t tell us anything about the distribution of the values between the largest and smallest For this reason we can also look at quartiles, deciles and/or percentiles
- 28. Quartiles, Percentiles and Deciles Quartiles : are points that subdivide the data into quarters Deciles : are points that subdivide the data into tenths Percentiles : are points that subdivide the data into hundredths
- 29. Quartiles First/lower quartile (Q 1 ) : is one quarter of the way through the data set when ordered from lowest to highest Second quartile (Q 2 ) = median Third/upper quartile (Q 3 ) : is three quarters of the way through the data set (in order)
- 30. Interquartile range = third quartile – first quartile The interquartile range is a better measure of dispersion than the range as it is not affected by ‘extreme’ values It indicates how densely the data is spread around the median
- 31. Semi-quartile range = Q 3 – Q 1 2 It is half of the interquartile range