Organising, Representing and Interpreting Data

Group discussions, Talk for Learning and experiential learning.
Learning Experience: Learners in mixed-ability groups discuss estimation measures of dispersion.
Activity 1:
- Learners in groups estimate quartiles of a given data.
- Learners recollect and construct cumulative frequency curves of a given data.
- Learners brainstorm and discover how to estimate quartiles from a cumulative frequency curve.
- Learner in their groups discover that;
- The lower quartile corresponds to the 25th percentile, i.e. 25% of the total frequency.
- The median corresponds to the 50th percentile, i.e. 50% of the total frequency.
- The upper quartile corresponds to the 75th percentile, i.e. 75% of the total frequency.
- The interquartile range = 𝑢 𝑞 - 𝑙 𝑞
- Learners solve practical examples and estimate the quartiles by plotting them on the cumulative frequency curve.
Example 1: The distribution of marks for 50 students in a test is given in the table. Draw a cumulative frequency curve to the distribution and use your graph to find the median, semi-interquartile range and pass mark if 30% of the students passed the test.
Marks Frequency 1-10 2 11-20 5 21-30 8 31-40 8 41-50 15 51-60 9 61-70 3
Activity 2:
- Learners in the collaborative groups work out measures of dispersion (Standard deviation).
- Learners in their collaborative groups conceptualise standard deviation and work out some examples.
- Learners recollect in their groups and share their ideas on standard deviation.
Expected Responses
- Standard deviation is a statistic that tells us the spread of data around the mean.
- Standard deviation tells the variation in the data
- The higher the standard deviation, the higher the spread
- If the data is close to the mean, the lower the standard deviation.
∑(0*0) !
- Learners in groups use the formula for standard deviation, which is y , to calculate the % standard deviation for raw data.
Example 1: The ages in years of 8 students are: 14, 14, 15, 15, 12, 11, 13, 10. Calculate the standard deviation
Solution: !,$!,$!.$!.$!)$!!$!"$!( Mean l𝑥n = = 13 1
𝑥 𝑥 − 𝑥 (𝑥 − 𝑥) ) 14 1 1 14 1 1 15 2 4 15 2 4 12 -1 1 11 -2 4 13 0 0 10 -3 9 Total 24
),
- ∴ 𝛿 = y 1 = 1.7
∑I0 !
- Learners in groups use the formula for standard deviation, which is y − (𝑥) ) , to calculate ∑I the standard deviation for ungrouped data
∑I0 ! ∑I0 )
- Learners in groups use the formula for standard deviation, which is y ∑I − i ∑I j , to
calculate the standard deviation for grouped data.
Activity 3: Learners in the collaborative groups work out measures of dispersion (Variance).
- Learners in their collaborative groups discuss variance and work out some examples.
- Learners establish that squaring the standard deviation gives the variance.
Example: Draw a frequency distribution table for the ages of some 50 citizens in a community below and calculate the standard deviation and variance of the distribution.
21 35 52 70 55 48 42 09 48 57 36 46 15 35 12 60 29 61 48 22 43 58 25 42 1 45 60 44 38 54 47 69 30 47 18 16 35 32 21 50 11 29 41 50 53 33 30 54 47 34
Midpoint, Marks 𝑥 ) 𝑓. (𝑓) 𝑓 𝑓𝑥 ) 𝑥 1 − 10 5.5 30.25 2 11 60.5 11 − 20 15.5 240.25 5 77.5 1201.25 21 − 30 25.5 650.25 8 204 5202 31 − 40 35.5 1260.25 8 284 10082 41 − 50 45.5 2070.2 15 682.5 31053.75 51 − 60 55.5 3080.25 9 499.5 27722.25 61 − 70 65.5 4290.25 3 196.5 12870.75 𝛴𝑓 𝛴𝑓𝑥 ) 𝛴 = 50 = 1955 = 88192.5
kI0 ) kI0 ! 𝑆 𝐷 = 𝜎 = y − i kI j kI !2.. ) 11!2).. =y − i .( j .( = √235.04 = 15.33 years 𝑉 = 𝜎 ) = 235.04 years
- Learners in groups discuss deviation taken from an assumed mean for grouped and ungrouped data.
- Learners recollect the idea of assumed mean and apply it in finding the actual mean and standard deviation.
- Learners discover the standard deviation given the assumed mean: ∑I<
- Actual mean= 𝐴 + ∑I
∑I< ! ∑I< )
- Standard deviation is 𝛿 = y ∑I − i ∑I j
Activity 4: Learners, in their collaborative work, research how to interpret measures of dispersion with respect to real data and present their findings in the classroom
Teaching and Learning Resources:
- SHS curriculum
- Research journals
- Internet or e-books
- Real-life examples
- Software (Excel, SPSS)
Assessment (1.4.1.AS.8). The document marks these depth-of-knowledge levels for this indicator: Level 3 Strategic reasoning; Level 4 Extended critical thinking and reasoning.