color":"green","style":"text-color","anchor":"q3_value"}]},"order":3,"title":"Find both quartiles","description":"Each half has five values, so its median is its third value. Correctly stating Q1 = 35 earns B1, and correctly stating Q3 = 60 earns B1."}]}],"generatedAt":"2026-07-20T16:26:28.385Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]},{"id":"9eab2c84-7464-4cbb-bc0f-99f426d66886","label":"b","content":"A value that is greater than $Q_3 + 1.5 \\times (Q_3 - Q_1)$ or smaller than $Q_1 - 1.5 \\times (Q_3 - Q_1)$ is defined as an outlier. Show that 110 is the only outlier.","markscheme":"- Upper limit: $60 + 1.5(60 - 35) = 97.5$ M1\n- Lower limit: $35 - 1.5(60 - 35) = -2.5$ M1\n- Conclusion: $110 > 97.5$, so 110 is an outlier. No other values are outside limits. A1","marks":3,"order":1,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Calculates the upper outlier boundary as 97.5.","code":"M"},{"id":"p2","marks":1,"description":"Shows 110 is above the boundary and no other values are outside.","code":"A"}],"finalAnswers":[],"generatedAt":"2026-05-12T02:14:06.551Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"outlier_limits","label":"Calculate the outlier limits","steps":[{"type":"text","order":0,"title":"Identify the quartiles","content":"The data are already in ascending order. With $11$ values, $Q_1$ is the $3$rd value and $Q_3$ is the $9$th value, so\n\n$$Q_1=35 \\quad\\text{and}\\quad Q_3=60.$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$U=\\htmlData{anchor=upperformula}{Q_3+1.5(Q_3-Q_1)}$"},{"latex":"$$U=\\htmlData{anchor=uppersub}{60+1.5(60-35)}$"},{"latex":"$$\\htmlData{anchor=upperanswer}{U=97.5}$"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"uppersub"},{"color":"green","style":"text-color-with-background","anchor":"upperanswer"}],"connections":[{"to":"uppersub","from":"upperformula","color":"sky","label":"substitute"},{"to":"upperanswer","from":"uppersub","color":"green","label":"calculate"}]},"order":1,"title":"Calculate the upper limit","description":"Substitute the quartiles into the upper outlier formula. This correct substitution earns the first M1 mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$L=\\htmlData{anchor=lowerformula}{Q_1-1.5(Q_3-Q_1)}$"},{"latex":"$$L=\\htmlData{anchor=lowersub}{35-1.5(60-35)}$"},{"latex":"$$\\htmlData{anchor=loweranswer}{L=-2.5}$"}],"highlights":[{"color":"amber","style":"text-color-with-background","anchor":"lowersub"},{"color":"green","style":"text-color-with-background","anchor":"loweranswer"}],"connections":[{"to":"lowersub","from":"lowerformula","color":"amber","label":"substitute"},{"to":"loweranswer","from":"lowersub","color":"green","label":"calculate"}]},"order":2,"title":"Calculate the lower limit","description":"Now substitute the same quartiles into the lower outlier formula. This earns the second M1 mark."},{"type":"text","order":3,"title":"Check all the values","content":"Compare the values with the two outlier limits:\n\n- $110>97.5$, so $110$ is above the upper limit.\n- The next-largest value is $77$, and $77<97.5$, so no other value is above the upper limit.\n- The smallest value is $17$, and $17>-2.5$, so no value is below the lower limit.\n\nTherefore, **$110$ is the only outlier**. This complete comparison and conclusion earns the A1 mark.","highlights":[{"text":"Show that 110 is the only outlier.","location":"specification"}]}]}],"generatedAt":"2026-07-20T16:30:17.067Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]},{"id":"43aaa711-e07e-417d-b497-3c3d344a1909","label":"c","content":"Draw a box plot for these data indicating clearly the position of the outlier.\n\n![](https://assets.mathsgenie.co.uk/booklet/2b6e9421bdb07aa9d90c6e430a13b50cc5c474148c0043da3969972cec25f203.jpg)","markscheme":"- Box with $Q_1, M, Q_3$ at 35, 53, 60 B1\n- Whiskers to 17 and 77 B1\n- Outlier marked with 'x' at 110 B1

Distance (nearest mile)	Frequency
0 – 9	4
10 – 14	19
15 – 18	41
19 – 20	26
21 – 25	9

Distance (nearest mile)	Frequency
0 – 9	4
10 – 14	19
15 – 18	41
19 – 20	26
21 – 25	9

Representations of Data

Question 4

Representations of Data Questions

Representations of Data

Question 4

Representations of Data Questions