Statistical measures
Two players on your school's Gaelic football team have the same mean score per game, but one scores the same every time and the other alternates between brilliant games and disasters. Which one would you pick for the county final? A single number does not tell the whole story: we need to know where the centre of the data is and how spread out it is.
Mean, median and mode
Before you calculate anything, ask who the data are about. The population is the whole group you want to know about; a sample is the part of it you actually ask or measure.
Change the data and see where the mean (pink line) and the median (green line) end up:
In a study on social media use, a group of teenagers was asked how many hours a day they spend watching videos on TikTok. The results obtained are:…
In a study on social media use, a group of teenagers was asked how many hours a day they spend watching videos on TikTok. The results obtained are:
$2, 3, 1, 4, 2, 3, 2, 5, 2, 1$
Identify the population of the study and calculate the mode of the hours of use.
Seven friends compare the number of unread messages they have on their messaging app. The data are as follows: $12, 15, 8, 20, 15, 14, 11$ Calculate the arithmetic mean…
Seven friends compare the number of unread messages they have on their messaging app. The data are as follows:
$12, 15, 8, 20, 15, 14, 11$
Calculate the arithmetic mean ($\bar{x}$) of the number of unread messages for this group.
A teacher organised the number of spelling mistakes made by their students in the last dictation in a frequency table: | Mistakes ($x_i$) | No. of students ($f_i$) | |…
A teacher organised the number of spelling mistakes made by their students in the last dictation in a frequency table:
| Mistakes ($x_i$) | No. of students ($f_i$) |
|---|---|
| $0$ | $8$ |
| $1$ | $12$ |
| $2$ | $6$ |
| $3$ | $4$ |
Calculate the mean number of mistakes in the class using the data from the table.
Aoife's marks in her first four Mathematics exams were $6.5$, $4$, $8$ and $7.5$. If she still has a fifth exam to take, what mark must she get for her final mean mark…
Aoife's marks in her first four Mathematics exams were $6.5$, $4$, $8$ and $7.5$. If she still has a fifth exam to take, what mark must she get for her final mean mark to be exactly $7$? Justify your answer by setting up an equation.
Quartiles and the box plot
Just as the median splits the ordered data into two halves, the quartiles mark positions around 25%, 50% and 75%:
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$Q_1$: first quartile, the 25% position.
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$Q_2$: the median (50%).
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$Q_3$: third quartile, the 75% position.
The box plot summarises everything in one diagram: the "whiskers" go from the minimum to the maximum and the box from $Q_1$ to $Q_3$, with a line at the median. It is a representation of the central 50%. With few data values or repeated values, these percentages should not be interpreted as exact counts of strictly smaller values.
Finding quartiles step by step
We will use this convention: put the data in order, find the median and take as $Q_1$ and $Q_3$ the medians of the lower and upper halves. If there is an odd number of data values, leave out the median when forming the halves. There are other conventions: state the one you have chosen and do not mix results from different methods.
The height of a group of 3rd Year students was measured and the data were ordered from smallest to largest. The values of the quartiles obtained are:…
The height of a group of 3rd Year students was measured and the data were ordered from smallest to largest. The values of the quartiles obtained are:
$Q_1 = 158 \text{ cm}, Q_2 = 165 \text{ cm}, Q_3 = 172 \text{ cm}$
- a) What is the value of the median?
- b) What percentage of students is $172 \text{ cm}$ tall or less?
Observe the following representation of a data set on a number line. The dots represent the number of pets owned by several neighbours: From the graph, write down the…
Observe the following representation of a data set on a number line. The dots represent the number of pets owned by several neighbours:
From the graph, write down the list of data and calculate the mode ($Mo$) and the range.
Spread: range and standard deviation
The range (maximum − minimum) is the simplest measure of spread, and it is part of the Junior Cycle.
Parameters from frequencies
When the raw data are a long list, first organise them into a frequency table: write each different value in order, make a tally mark for each time it appears, then count the tallies to get the frequency $f_i$. Check that the frequencies add up to $N$, the number of data values.
A group of 20 teenagers recorded how many times they went to the cinema in the last six months. The data obtained are as follows:…
A group of 20 teenagers recorded how many times they went to the cinema in the last six months. The data obtained are as follows:
$2, 3, 0, 1, 5, 3, 2, 1, 0, 0, 2, 4, 3, 2, 1, 1, 6, 2, 0, 3$
Organise these data into a frequency table and calculate the measures of central tendency: the arithmetic mean ($\bar{x}$), the median ($Me$) and the mode ($Mo$). Which of them do you consider best represents the group?
With frequencies, $s^2 = \dfrac{\sum f_i(x_i-\bar{x})^2}{N}$. Do not calculate the mean or the variance of the column of values without taking into account how many times each one appears.
Higher Level only: the mean of grouped data.
Comparing and deciding
Two teams take, in minutes, A: 8, 10, 12 and B: 2, 10, 18. Both have a mean of 10, but their standard deviations are approximately 1.63 and 6.53. If you need predictable times, A is more consistent. The CV is approximately 0.163 (16.3%) for A and 0.653 (65.3%) for B. Lower spread does not mean one team is better for every purpose: explain what you need to compare.
During a training week, an athlete recorded the kilometres she ran each day to prepare for a race: $5, 8, 10, 12, 15, 18, 21$ Find the median ($Me$) of the distance…
During a training week, an athlete recorded the kilometres she ran each day to prepare for a race:
$5, 8, 10, 12, 15, 18, 21$
Find the median ($Me$) of the distance covered. What does this value mean in relation to the total number of days trained?
Two esports teams recorded their scores in the last 5 matches. Team A: $85, 88, 90, 92, 95$ Team B: $60, 75, 90, 105, 120$ Both teams have the same mean, $\bar{x} = 90$.…
Two esports teams recorded their scores in the last 5 matches.
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Team A: $85, 88, 90, 92, 95$
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Team B: $60, 75, 90, 105, 120$
Both teams have the same mean, $\bar{x} = 90$. Without performing complex standard deviation calculations, look at the data and explain which of the two teams is more consistent (has less dispersion) and why.
Two messaging apps, "App A" and "App B", are competing to be the most stable. The response time (in milliseconds) was measured over 10 different attempts, obtaining the…
Two messaging apps, "App A" and "App B", are competing to be the most stable. The response time (in milliseconds) was measured over 10 different attempts, obtaining the following results:
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App A: $\bar{x} = 120$ ms; $\sigma = 15$ ms
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App B: $\bar{x} = 125$ ms; $\sigma = 5$ ms
Calculate the coefficient of variation ($CV$) for both applications and explain with reasons which of the two is more "reliable" or consistent in its performance for a user.
Look at the following two data sets, which represent the points scored by two basketball players in their last 5 games: Player A: $18, 20, 22, 19, 21$ Player B:…
Look at the following two data sets, which represent the points scored by two basketball players in their last 5 games:
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Player A: $18, 20, 22, 19, 21$
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Player B: $10, 30, 5, 25, 30$
In the figure you have the same data represented on a number line (each dot is a game).
Without doing complex variance calculations, look at the data and answer:
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Do they have the same arithmetic mean?
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Which of the two players is more "consistent" in their scoring?
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If you had to choose one of them to play the last minute of a final, which one would you choose and why? Use the concepts of mean and dispersion in your answer.
Back to the start: You would pick the consistent player (low standard deviation) if you want reliable scores over a league season, or the inconsistent one (high standard deviation) for a risky bet in a final where only an extraordinary score will do.