Statistics flashcards

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Define mean, median, mode and range.
Mean: total \(\div\) number of values. Median: middle value when ordered. Mode: most frequent value. Range: largest minus smallest.
Find the mean, median, mode and range of 3, 7, 7, 8, 10.
Mean \(=\frac{35}{5}=7\). Median \(=7\). Mode \(=7\). Range \(=10-3=7\).
How do you find the median of an even number of values?
Order the data and take the mean of the two middle values. E.g. 2, 4, 6, 10 has median \(\frac{4+6}{2}=5\).
When is the median a better average than the mean?
When there are extreme values (outliers) or skewed data, because the median is not pulled towards them. The mean uses every value and is affected by them.
When is the mode the most useful average?
For non-numerical (categorical) data, such as most popular colour, or when you want the most common value, e.g. the most common shoe size.
How do you find the mean from a frequency table?
Multiply each value by its frequency, add these to get \(\sum fx\), then divide by the total frequency \(\sum f\).
A table shows scores 1, 2, 3 with frequencies 4, 3, 3. Find the mean.
\(\frac{1(4)+2(3)+3(3)}{10}=\frac{19}{10}=1.9\).
How do you estimate the mean from grouped data?
Use the midpoint of each class as its value. Mean \(\approx\frac{\sum f\times\text{midpoint}}{\sum f}\). It is an estimate because individual values are unknown.
Estimate the mean: classes 0–10 (freq 4), 10–20 (freq 6), 20–30 (freq 10).
Midpoints 5, 15, 25: \(\frac{20+90+250}{20}=\frac{360}{20}=18\).
How do you identify the modal class and the class containing the median?
The modal class has the highest frequency (equal class widths). The median class contains the \(\frac{n+1}{2}\)th value, found from running totals.
(Extended) What is a cumulative frequency table and how do you plot its graph?
Add up frequencies as a running total. Plot each total against the upper class boundary, then join the points with a smooth curve.
(Extended) How do you read the median and quartiles from a cumulative frequency curve?
For total \(n\): median at \(\frac n2\), lower quartile at \(\frac n4\), upper quartile at \(\frac{3n}{4}\) on the vertical axis. Read across to the curve, then down to the value.
(Extended) What is the interquartile range and what does it measure?
IQR \(=Q_3-Q_1\). It is the spread of the middle 50% of the data, so it is not affected by outliers, unlike the range.
(Extended) From a cumulative frequency curve with \(n=80\), how many values are above 45 if the curve reads 62 at 45?
\(80-62=18\) values are above 45.
What does a box-and-whisker plot show?
The minimum, lower quartile, median, upper quartile and maximum. The box covers the IQR and whiskers extend to the extremes.
How can you compare two data sets using averages and spread?
Compare a measure of average (median or mean) and a measure of spread (range or IQR). Write the comparison in context: 'On average A is higher, and B is more consistent'.
(Extended) How do you draw a histogram with unequal class widths?
Plot frequency density on the vertical axis, \(\text{frequency density}=\frac{\text{frequency}}{\text{class width}}\). The area of each bar is proportional to the frequency.
(Extended) A histogram bar has width 5 and height 3.2. Find the frequency.
Frequency \(=\) frequency density \(\times\) class width \(=3.2\times5=16\).
(Extended) A class 20–40 has frequency 30. Find the frequency density.
\(\frac{30}{20}=1.5\).
What are the types of correlation shown in a scatter diagram?
Positive: as one variable increases the other increases. Negative: one increases as the other decreases. Zero: no linear relationship.
What is a line of best fit and how do you use it?
A straight line drawn through the points with about equal numbers on each side, passing through the mean point. Use it to estimate one value from the other, within the data range.
Why is it unreliable to use a line of best fit to predict far outside the plotted data?
This is extrapolation: the trend may not continue beyond the data. Interpolation, within the range, is more reliable.
Does correlation prove that one variable causes the other?
No. Two variables can be correlated through a third factor or by chance. E.g. ice cream sales and sunburn both rise in hot weather.
How do you draw an accurate pie chart angle for a category?
Angle \(=\frac{\text{frequency}}{\text{total}}\times360^\circ\). E.g. 12 out of 60 gives \(72^\circ\).
A pie chart has a sector of \(90^\circ\) representing 18 people. How many people in total?
\(90^\circ\) is \(\frac14\) of the circle, so total \(=18\times4=72\).
What is the difference between discrete and continuous data?
Discrete data can take only separate values, usually counted (shoe size, number of siblings). Continuous data can take any value in a range, usually measured (height, time).
What makes a good sample in a survey?
It should be large enough and unbiased, representing the whole population (e.g. chosen at random). A biased sample can give misleading conclusions.
The mean of 5 numbers is 8. A sixth number is added and the mean becomes 9. Find the sixth number.
New total \(=6\times9=54\) and old total \(=5\times8=40\). The sixth number is \(54-40=14\).
How do you read a stem-and-leaf diagram and what must it include?
Stems are the leading digits, leaves are the last digits, in order. It needs a key, e.g. '3 | 4 means 34'. The median and mode can be read from the ordered leaves.

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