Statistics and probability flashcards

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Define the mean, median, mode and range, and when is the mean a poor average?
Mean: total ÷ number of values. Median: middle value in order. Mode: most common. Range: largest − smallest. An extreme value (outlier) distorts the mean, so the median may be better.
Find the mean, median, mode and range of 3, 7, 7, 9, 14.
Mean \(=40\div5=8\); median \(=7\); mode \(=7\); range \(=14-3=11\). Put the data in order before choosing the median.
How do you find the mean from a frequency table? Use x = 1, 2, 3 with f = 4, 5, 1.
Mean \(=\dfrac{\sum fx}{\sum f}\). \(\dfrac{4+10+3}{10}=1.7\). Multiply each value by its frequency; do not just add the values.
How do you estimate the mean from grouped data? Classes 0 < x ≤ 10 (f = 4) and 10 < x ≤ 20 (f = 6).
Use the class midpoints (5 and 15): \(\dfrac{4\times5+6\times15}{10}=\dfrac{110}{10}=11\). It is an estimate because the exact values are unknown.
In grouped data, what are the modal class and the median class?
The modal class is the class with the highest frequency (equal class widths). The median class is the class that contains the middle value, found from the cumulative frequency.
(Higher) How do you draw a cumulative frequency graph, and where do you read the median and quartiles?
Plot the running total against the upper class boundary and join with a smooth curve. For \(n\) values read the median at \(\tfrac n2\), lower quartile at \(\tfrac n4\) and upper quartile at \(\tfrac{3n}4\).
What is the interquartile range, and what does it tell you?
IQR \(=\) upper quartile \(-\) lower quartile; for example \(41-24=17\). It measures the spread of the middle half of the data and is not affected by outliers.
(Higher) Which five values are needed to draw a box plot?
Minimum, lower quartile, median, upper quartile and maximum. The box runs from the lower to the upper quartile with a line at the median; whiskers go to the minimum and maximum.
How should you compare two distributions in an answer?
Give one comparison of averages (median or mean) and one of spread (range or IQR), each written in the context of the question and with the values quoted.
(Higher) How is a histogram drawn when the class widths differ?
Plot frequency density on the vertical axis, where frequency density \(=\dfrac{\text{frequency}}{\text{class width}}\). The area of each bar is proportional to the frequency, and the bars have no gaps.
(Higher) A histogram bar has width 5 and height 3.2. What is the frequency? Another class, 10–30, has frequency 24; find its density.
Frequency \(=\) area \(=5\times3.2=16\). Density \(=24\div20=1.2\).
(Higher) Common error: when plotting a cumulative frequency graph, which value is used on the x-axis, and for a frequency polygon?
Cumulative frequency uses the upper class boundary, because the total is reached at the end of the class. A frequency polygon uses the class midpoints.
What do positive, negative and zero correlation mean on a scatter graph?
Positive: as one variable increases, so does the other. Negative: as one increases, the other decreases. Zero: no linear relationship between them.
How do you use a line of best fit, and why is extrapolation risky?
Draw a straight line with points balanced on each side, passing near the mean point, then read off estimates. Using it outside the data range (extrapolation) assumes the trend continues, which may be false.
State the basic rules of probability.
\(P(A)=\dfrac{\text{favourable outcomes}}{\text{total outcomes}}\), values lie between 0 and 1, probabilities of all outcomes add to 1, and \(P(\text{not }A)=1-P(A)\). If \(P(\text{rain})=0.35\), \(P(\text{no rain})=0.65\).
How do you use relative frequency to estimate probability and expected outcomes?
Estimate \(=\dfrac{\text{number of successes}}{\text{number of trials}}\). Red 18 times in 60 spins gives 0.3. In 200 spins, expect \(0.3\times200=60\) reds. More trials give a better estimate.
What is the rule for mutually exclusive events? Find P(A or B) when P(A) = 0.2 and P(B) = 0.35.
Events that cannot happen together: \(P(A\text{ or }B)=P(A)+P(B)=0.2+0.35=0.55\).
What is the rule for independent events? Find P(A and B) when P(A) = 0.6 and P(B) = 0.5.
Events where one does not affect the other: \(P(A\text{ and }B)=P(A)\times P(B)=0.6\times0.5=0.3\).
A bag has 3 red and 2 blue counters. Two are taken with replacement. Find P(two red) and P(one of each).
\(P(RR)=\tfrac35\times\tfrac35=\tfrac9{25}\). One of each: \(RB\) or \(BR\), so \(2\times\tfrac35\times\tfrac25=\tfrac{12}{25}\). Multiply along branches, add between branches.
Now take the two counters without replacement. Find P(two red).
The second probability changes because only 4 counters remain: \(\tfrac35\times\tfrac24=\tfrac3{10}\).
(Higher) What is conditional probability? A bag has 4 red and 3 blue counters; two are taken without replacement. Given the first is red, find P(second red).
A probability that depends on something already known. After one red is removed, 3 red remain among 6 counters: \(\dfrac36=\dfrac12\).
In a Venn diagram with 40 people, 6 are in both A and B. How do you find P(A and B), and what does 'at least one' suggest?
\(P(A\cap B)=\dfrac6{40}=0.15\). For 'at least one', it is often easier to use \(1-P(\text{none})\): three coin flips give \(1-\tfrac18=\tfrac78\) for at least one head.
How do you use a sample space diagram for two dice? Find P(total 7).
List all \(6\times6=36\) equally likely outcomes. Six give a total of 7, so \(P=\dfrac6{36}=\dfrac16\).
What makes a sample random and why does bias matter?
In a random sample every member of the population has an equal chance of selection. A biased sample (for example only asking people at a gym) does not represent the population, and a larger sample is generally more reliable.
How do you draw a pie chart? Find the angle for 15 people out of 60.
Angle \(=\dfrac{\text{frequency}}{\text{total}}\times360^\circ\). \(\dfrac{15}{60}\times360=90^\circ\).
Distinguish discrete, continuous, qualitative and quantitative data.
Qualitative data is described in words (colour). Quantitative data is numerical: discrete if it takes separate values (shoe size, number of goals), continuous if it can take any value in a range (height, time).
What are the rules for drawing a stem-and-leaf diagram?
Put the leaves in order, align them in columns and give a key such as 3 | 4 = 34. The median is the middle leaf when the data are counted in order.
How is a two-way table used?
It shows two categories at once (for example year group and transport). Fill in missing values using the row and column totals; probabilities use the relevant row or column total as the denominator.

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