Wynberg girls high-Jade Gibson-maths-data analysis statistics

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

… 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)

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

Arithmetic methods: involve working out: Measures of central tendency – mean median and mode Measure of dispersion – range, percentiles, quartiles and the interquartile range

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

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#

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.

“ 86 % of statistics are made up on the spot and the remaining 24% are flawed”

Measures of central tendency Mean, mode and median “Averages”

… 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

… 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

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

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)

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

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

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

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

… 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

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

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

… 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

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

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

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

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)

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

Semi-quartile range = Q 3 – Q 1 2 It is half of the interquartile range

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