Flashcards · 11 cards

Dispersion & cumulative frequency

Say the answer to yourself, then check. Cards you know come back less and less often; cards you don't come back tomorrow.

Know it
The range of a set of data?
Answer
Largest value − smallest value.
From the lesson: Range, quartiles and standard deviation
Rule
How do you find $Q_1$ and $Q_3$ of a list of $n$ values?
Answer
Sort the list. $Q_1$ is at position $\frac n4$ and $Q_3$ at $\frac{3n}{4}$ . A position ending in .5: halfway between the two values either side. Ending in .25 or .75: round up to the next position.
Quartiles of 11 valuesQ₁ = 5, median = 8, Q₃ = 12, so the interquartile range is 7
From the lesson: Range, quartiles and standard deviation
Know it
The interquartile range and the semi-interquartile range?
Answer
Interquartile range $= Q_3 - Q_1$ , the spread of the middle half. The semi-interquartile range is half of it.
From the lesson: Range, quartiles and standard deviation
Rule
The mean deviation?
Answer
$\dfrac{\sum |x - \bar x|}{n}$ : every distance from the mean counted as positive.
Distances from the meanMean deviation = (4 + 1 + 0 + 2 + 3) ÷ 5 = 2
From the lesson: Range, quartiles and standard deviation
Know it
The variance and standard deviation of a list?
Answer
Variance $= \dfrac{\sum (x - \bar x)^2}{n}$ , dividing by $n$ . The standard deviation is its square root.
From the lesson: Range, quartiles and standard deviation
Know it
The standard deviation from a frequency table?
Answer
$\sqrt{\dfrac{\sum f(x - \bar x)^2}{\sum f}}$ : square each deviation first, then multiply by $f$ . Or $\sqrt{\dfrac{\sum fx^2}{\sum f} - \bar x^2}$ .
From the lesson: Range, quartiles and standard deviation
Rule
Every value is increased by 10, or doubled. What happens to the mean and the standard deviation?
Answer
Adding 10: the mean goes up by 10, the standard deviation is unchanged. Doubling: both double.
Moving and stretchingAdding 10 keeps the spread; doubling doubles it
From the lesson: Range, quartiles and standard deviation
Know it
Drawing a cumulative frequency curve: where do you plot each point?
Answer
At the upper class boundary (9.5, 19.5, …), against the running total. Join the points with a smooth curve.
From the lesson: The cumulative frequency curve
Rule
How do you read the median and quartiles from the curve?
Answer
Go across from $\frac14 N$ , $\frac12 N$ and $\frac34 N$ to the curve, then down to the marks.
Reading the curveAcross from ¼N, ½N and ¾N, then down
From the lesson: The cumulative frequency curve
Know it
Using the curve, how many scored at least 66?
Answer
The curve gives the number below a mark. Read the number below 66, then take it from the total.
From the lesson: The cumulative frequency curve
Which method?
WAEC 2020 · Paper 2 · Q10

Marks (%) 0–9 10–19 20–29 30–39 40–49 50–59 60–69 70–79 80–89 90–99

Frequency 7 11 17 20 29 34 30 25 21 6

The table shows the distribution of marks obtained by students in an examination.
Construct a cumulative frequency table for the distribution.
Draw the cumulative frequency curve for the distribution.
Using the curve, find, correct to one decimal place, the: (i) median mark; (ii) lowest mark for distinction if $5\%$ of the students passed with distinction.
The top 5% get a distinction. Where do you read the curve?
Answer
The curve counts from the bottom, so the top 5% begin at 95% of the students: go across from $0.95 \times 200 = 190$ , then down.
From the lesson: The cumulative frequency curve Try the question