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Flashcards · 9260 · Topic 4

Statistics and probability flashcards

35 cards on Topic 4, subtopics 3.4.1 to 3.4.3. Pick a subtopic or study them all.

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3.4.1 Presentation and analysis

Qualitative, discrete and continuous data
Qualitative: described in words (colour). Discrete: numerical, separate values, counted (shoe size). Continuous: numerical, any value in a range, measured (time, mass).
Classes 30 ≤ x < 40 and 40 ≤ x < 50: which class holds 40?
The second class, 40 ≤ x < 50. Class intervals must not overlap and must leave no gaps.
x = 10, 11, 12, 13 with frequencies 3, 7, 6, 4. Find the mean and median.
Σfx = 30 + 77 + 72 + 52 = 231; Σf = 20; mean = 231 ÷ 20 = 11.55. Median at position 10.5: the 10th value is 11 and the 11th is 12, so the median is 11.5.
How do you find the class that contains the median of grouped data?
Work out n ÷ 2, then add frequencies down the table until the running total first reaches or passes it. You can give only the class, not an exact median.
8 items have mean 5 and 2 items have mean 10. Find the mean of all 10.
Total = 8 × 5 + 2 × 10 = 60. Mean = 60 ÷ 10 = 6.
Find the IQR of 4, 6, 9, 12, 15, 18, 25Extension
n + 1 = 8. Lower quartile at position 2: 6. Upper quartile at position 6: 18. IQR = 18 − 6 = 12
Pie chart: 45 pupils choose art 9, drama 14, music 22. Find the angles.
One pupil = 360° ÷ 45 = 8°. Art 72°, drama 112°, music 176° (total 360°).
Frequency polygon
Plot each frequency above its class midpoint and join the points with straight lines. Midpoint = (lower boundary + upper boundary) ÷ 2.
Histogram with unequal class widthsExtension
Bar height = frequency density = frequency ÷ class width. Bars touch, and area represents frequency (frequency = density × width). Example: 44 in 10 < x ≤ 20 gives density 4.4.
Cumulative frequency diagram: where are points plotted, and where do you read the quartiles?Extension
Plot each running total at the upper end of its class, plus (start, 0), and draw a smooth curve. Read across from n ÷ 4, n ÷ 2 and 3n ÷ 4.
Box plot: what five values does it show?Extension
Smallest value, lower quartile, median, upper quartile and largest value. The box runs between the quartiles and is cut at the median.
Bar chart and histogram: the difference
A bar chart has gaps and shows categories or discrete values. A histogram has no gaps and a continuous scale.

3.4.2 Interpretation

What does a full comparison of two distributions include?
An average for each set with its value, a measure of spread for each with its value, a sentence saying which is higher or more consistent in context, and a reason or caution.
Scores P: 6, 8, 9, 10, 12 and Q: 2, 9, 9, 10, 15. Compare them.
Both have mean 9 and median 9, so typical performance is equal. Ranges: P 6, Q 13. P is more consistent.
Signs that a graph may mislead
The vertical axis does not start at zero; the scale is uneven or bars have different widths; a 3-D or stretched picture; a very small sample or no source.
Positive, negative and no correlation
Positive: points rise left to right. Negative: points fall left to right. None: no pattern. State strength too, for example "weak positive correlation".
How do you draw a line of best fit by eye?
Find the mean point (mean of x, mean of y). Draw one straight ruled line through or very close to it, following the trend with points balanced above and below. Leave out outliers.
Interpolate and extrapolate
Interpolate: estimate inside the range of the data (fairly reliable). Extrapolate: estimate outside the range of the data (not reliable).
Does correlation mean that one variable causes the other?
No. Correlation shows a relationship; a third factor may affect both.
Outlier
A point well away from the general pattern.
A line of best fit passes through (0, 10) and (5, 35). Find its equation and estimate y at x = 3.
Gradient = 25 ÷ 5 = 5, intercept 10: y = 5x + 10. At x = 3: y = 25.

3.4.3 Probability

The probability scale
0 impossible, 0.5 evens, 1 certain. Probabilities are fractions, decimals or percentages, never ratios, and lie from 0 to 1.
Fair and biased
Fair (unbiased): all outcomes equally likely. Biased: some outcomes favoured.
Relative frequency: a pin lands point up 74 times in 200 drops. How many in 1500 drops?
Relative frequency = 74/200 = 0.37 (an estimate of the probability). Expected frequency = 0.37 × 1500 = 555.
Expected frequency: a fair spinner has 8 equal sectors, 3 shaded. Spun 240 times?
P(shaded) = 3/8. Expected = probability × trials = 3/8 × 240 = 90.
Why do more trials usually give a better estimate?
More trials generally give a relative frequency closer to the true probability (likely, not certain). A large sample taken in a biased way is still a poor sample.
A coin has landed heads five times in a row. What is P(heads) next time?
Still 1/2. Past results do not change the probability of the next trial.
Mutually exclusive and exhaustive
Mutually exclusive: the events cannot both happen (no overlap). Exhaustive: at least one of the outcomes must happen (no gap).
P(win) = 0.35, P(draw) = 0.2 and P(lose) = y. Find y.
The outcomes are mutually exclusive and exhaustive, so they sum to 1: 0.35 + 0.2 + y = 1, so y = 0.45.
Of 30 pupils, 14 have a dog, 11 a cat and 5 both. Find P(neither).
Dog only 9, both 5, cat only 6: 20 in the circles. Neither = 30 − 20 = 10, so P(neither) = 10/30 = 1/3.
Two independent machines run with probabilities 0.8 and 0.7. Find P(at least one runs).Extension
P(neither) = 0.2 × 0.3 = 0.06. P(at least one) = 1 − 0.06 = 0.94. For independent events, P(A ∩ B) = P(A) × P(B).
A bag has 3 red and 8 blue counters. Two are taken without replacement. Find P(same colour).Extension
P(red, red) = 3/11 × 2/10 = 6/110. P(blue, blue) = 8/11 × 7/10 = 56/110. P(same) = 62/110 = 31/55.
Frost has probability 0.4. With frost a car fails to start with probability 0.2; without, 0.05. Find P(frost | fails).Extension
P(frost and fails) = 0.08; P(no frost and fails) = 0.6 × 0.05 = 0.03; P(fails) = 0.11. P(frost | fails) = 0.08 ÷ 0.11 = 8/11.
Using a two-way table for conditional probabilityExtension
It is ideal for conditional probability: read the given row or column total as the new total. Example: 9 of 24 juniors play chess, so P(chess | junior) = 9/24 = 3/8.
A player scores each shot with probability 0.6, independently. Find P(exactly two of three shots).Extension
One arrangement (score, score, miss): 0.6 × 0.6 × 0.4 = 0.144. There are 3 arrangements, so P = 3 × 0.144 = 0.432.

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