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Aims: To be able to find the smallest & largest values along with the median, quartiles and IQR To be able to draw a box and whisker plot To be able to interpret such a diagram and look for skewness. Representing data Lesson 2
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Quartiles When using the median to compare data it makes sense to consider what data values fall at other significant positions through the data. We often produce a 5 number summary of the data comprising of values at 0% (first); 25% ( 1 / 4 way through); 50% (Median); 75% ( 3 / 4 way through); 100% (Last) when the data is placed in size order (This can be found on your GDC) These 25% and 75% values are called the “Lower Quartile” and “Upper Quartile” respectively.
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Calculating Quartiles If n is even…order the data 3, 7, 15, 19, 20, 22 split the data in half… 3,7,15 19, 20, 22 Identify the medians of each half Lower Quartile=7, Upper Quartile = 20 If n is not even the median of the data is not counted leaving two halves for which a median can be identified. 1,3,4,6,7,8,10,13,20 median of data is 7 1,3,4,6 8,10,13,20 then find medians of the halves lower quartile = 3.5 and upper quartile = 11.5
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Interquartile Range To find the Interquartile Range you simply subtract the Lower Quartile from the Upper Quartile. So for the last two examples the IQR would be 20-7=13 then 11.5-3.5=8
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For This Data Find The... Amount smoked on weekdays by 15 random male smokers 0, 8, 10, 15, 15, 20, 20, 20, 20, 20, 30, 30, 30, 30, 40 Median: Lower Quartile: Upper Quartile: IQR: Amount smoked on weekdays by 12 random female smokers 2, 2, 5, 5, 10, 12, 15, 15, 20, 20, 25, 30 Median: Lower Quartile: Upper Quartile: IQR:
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A set of data can be summarised using 5 key statistics: Quartiles and box plots the median value (denoted Q 2 ) – this is the middle number once the data has been written in order. If there are n numbers in order, the median lies in position ½ ( n + 1). the lower quartile ( Q 1 ) – this value lies one quarter of the way through the ordered data; the upper quartile ( Q 3 ) – this lies three quarters of the way through the distribution. the smallest value and the largest value. In the exercise this is to as the five-referred number summaries
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These five numbers can be shown on a simple diagram known as a box-and-whisker plot (or box plot): Smallest value Q1Q1 Q2Q2 Q3Q3 Largest value Note: The box width is the inter-quartile range. Inter-quartile range = Q 3 – Q 1 Quartiles and box plots The inter-quartile range is a measure of spread. The semi-inter-quartile range = ½ ( Q 3 – Q 1 ).
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Example: The (ordered) ages of 15 brides marrying at a registry office one month in 1991 were: 18, 20, 20, 22, 23, 23, 25, 26, 29, 30, 32, 34, 38, 44, 53 The median is the ½(15 + 1) = 8 th number. So, Q 2 =. The lower quartile is the median of the numbers below Q 2, So, Q 1 = 22 The upper quartile is the median of the numbers above Q 2, So, Q 3 = 34. The smallest and largest numbers are 18 and 53. Quartiles and box plots 26
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The (ordered) ages of 12 brides marrying at the registry office in the same month in 2005 were: 21, 24, 25, 25, 27, 28, 31, 34, 37, 43, 47, 61 Q 2 is half-way between the 6 th and 7 th numbers: Q 2 = 29.5. Q 1 is the median of the smallest 6 numbers: Q 1 = 25 Q 3 is the median of the highest 6 numbers: Q 3 = 40. The smallest and highest numbers are 21 and 61. Quartiles and box plots
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We can use the box plots to compare the two distributions. The median values show that the brides in 1991 were generally younger than in 2005. The inter-quartile range was larger in 2005 meaning that that there was greater variation in the ages of brides in 2005. Note: When asked to compare data, always write your comparisons in the context of the question. Quartiles and box plots A box plot to compare the ages of brides in 1991 and 2005 It is important that the two box plots are drawn on the same scale.
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Box plots are useful because they make comparing the location, spread and the shape of distributions easy. A distribution is roughly symmetrical if Q 2 – Q 1 ≈ Q 3 – Q 2 A distribution is positively skewed if Q 2 – Q 1 < Q 3 – Q 2 A distribution is negatively skewed if Q 2 – Q 1 > Q 3 – Q 2 Shapes of distributions
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Order starting with smallest:
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Card match Sets of 4 frequency, cumulative frequency, box plot, descriptions In table groups
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Examination style question: A survey was carried out into the speed of traffic (in mph) on a main road at two times: 8 a.m. and 11 a.m. The speeds of 25 cars were recorded at each time and displayed in a stem-and-leaf diagram: a)Find the median and the inter-quartile range for the traffic speeds at both 8 a.m. and 11 a.m. A stem-and-leaf diagram to show vehicle speed on a main road Examination style question The L.Q. would be the median of the smallest 12 values. The U.Q. would be the median of the largest 12 values 8 a.m. 11 a.m. Q1Q1 Q3Q3 IQR 34 15 6049 42 18
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The box plots show that traffic speed is generally slower at 8 a.m. than at 11 a.m. The inter-quartile ranges show that there is greater variation in the traffic speed at 11 a.m. than at 8 a.m. Notice that the speeds at 8 a.m. have a negative skew, whilst the speeds at 11 a.m. are roughly symmetrically distributed. 8 a.m.11 a.m. Q 2 43 51 Q 1 34 42 Q 3 49 60 IQR 15 18 Examination style question A box plot comparing vehicle speed at 8 a.m. and 11 a.m. Do exercise 3A page 47, questions 1, 3, 7, 8 and 10. Do Miscellaneous exercise page 58, question 8. b)Draw box plots for the two sets of data and compare the speeds of the traffic at the two times.
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