Support for English, maths and ESOL Module 7: Developing the personal maths skills of teachers and assessors Master class 4: Handling data

Session aim To review and extend participants’ personal mathematics relating to statistics and probability To apply concepts of statistics and probability to solve problems

Outcomes By the end of the session participants will have: Reviewed their understanding of mean, median, mode and range, and their relationship to frequency distributions Examined and understood different measures of spread Explored features of grouped and cumulative frequency distributions and their uses Used scatter diagrams to investigate correlation, and make projections Reviewed key concepts of probability, including combined probabilities Calculated probabilities from frequency data Applied understanding of statistics and probability to solve practical problems

What’s the difference?

Measures of dispersion Range Interquartile range Variance Standard deviation

Box and whisker plots Group A Group B Group A: mean = 3.5, median = 3.5, mode = 3 & 4, range = 3 Group B: mean = 3.5, median = 3.5, mode = 4, range = 5

Standard deviation Use the population standard deviation (σ) when data is relates to the entire population:   The standard deviation of a sample of the population (s) can be estimated using: Note: xi represents the individual values of x; μ is the mean of all the values of x N is the number of values

Frequency diagram

Grouped frequency distribution Score 0-10 11-20 21-30 31-40 41-50 51-60 61-70 Frequency 10 60 230 420 250 28 2 450 400 350 300 250 200 150 100 50

Cumulative frequency table Score 0-10 11-20 21-30 31-40 41-50 51-60 61-70 Frequency 10 60 230 420 250 28 2 Score 10 20 30 40 50 60 70 Cumulative frequency 10 70 300 720 970 998 1000

Cumulative frequency graph Score 10 20 30 40 50 60 70 Cumulative frequency 10 70 300 720 970 998 1000

Box and whisker plot Lower quartile Upper quartile Median

What’s misleading? Score 0-10 11-20 21-30 31-50 51-100 101-150 Frequency 20 38 42 18 6 1 Frequency 45 40 35 30 25 20 15 10 5 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 Score

Frequencies proportional to area of bars Score 0-10 11-20 21-30 31-50 51-100 101-150 Frequency 20 38 42 18 6 1 Frequency 45 40 35 30 25 20 15 10 5 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 Score

Correlation

Negative correlation

P(a) = No. successful outcomes Total possible outcomes Probability P(a) = No. successful outcomes Total possible outcomes

Combined probabilities When two coins are tossed there are three possible outcomes: Two heads One head No heads The probability of two heads is therefore ⅓

Train punctuality data Scheduled services running in the last month: On time 84% Up to 15 min. late 7% 15-30 min. late 4% More than 30 min. late 2% Cancelled 3%

Probability and frequency data Score 0-10 11-20 21-30 31-40 41-50 51-60 61-70 Frequency 10 60 230 420 250 28 2 450 400 350 300 250 200 150 100 50 Frequency 0 10 20 30 40 50 60 70 80 90 100 Score