Measures of central tendency and dispersion
Two mobile phones have the same mean battery life. Are they equally reliable? A single number does not tell the whole story: we need to know where the "centre" of the data is and whether the values are close together or widely spread.
Mean, median and mode
Change the data and see where the mean and the median end up:
Categorical data
Mean and median only make sense for numerical data. For categorical data (such as favourite colour or brand of phone) you can only find the mode, because the values cannot be added or put in a meaningful order. Numerical data are discrete (counted, e.g. number of likes) or continuous (measured, e.g. time in minutes).
Finding a missing value from the mean
Since $\bar{x} = \dfrac{\text{total}}{N}$, we have $\text{total} = \bar{x} \times N$. To find a missing value, work out the total needed and subtract the total you already have.
To carry out a study on the consumption habits of teenagers on a social network, the profiles of $50$ students from a secondary school were analysed. Classify the…
To carry out a study on the consumption habits of teenagers on a social network, the profiles of $50$ students from a secondary school were analysed. Classify the following statistical variables obtained in the study as qualitative or quantitative (indicating whether they are discrete or continuous):
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a) Brand of mobile phone used.
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b) Number of "likes" received on the latest post.
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c) Total time (in minutes) spent using the app during the day.
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d) Main topic of the videos watched (sports, comedy, music, etc.).
A group of friends recorded the number of unread WhatsApp messages they have at the end of the day. The data obtained is: $$12, 15, 12, 10, 18, 12, 14, 20, 12, 15$$…
A group of friends recorded the number of unread WhatsApp messages they have at the end of the day. The data obtained is:
$$12, 15, 12, 10, 18, 12, 14, 20, 12, 15$$Calculate the mode of this distribution and briefly explain what this value represents in the context of the exercise.
A young person saves different amounts of money weekly. Over the last $5$ weeks, his savings have been: $$\text{€}\ 5, \text{€}\ 8, \text{€}\ 6, \text{€}\ 7, \text{€}\…
A young person saves different amounts of money weekly. Over the last $5$ weeks, his savings have been:
$$\text{€}\ 5, \text{€}\ 8, \text{€}\ 6, \text{€}\ 7, \text{€}\ 54$$-
a) Calculate the mean of his savings.
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b) Calculate the median.
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c) Which of the two measures do you think best represents the "normal" savings of this young person? Justify your answer by mentioning the effect of the value $\text{€}\ 54$.
Aisha has taken 4 Mathematics exams this term and her marks were: $6.5$; $8.2$; $7.4$ and $9.1$. a) What is her current mean mark? b) If there is still one final exam…
Aisha has taken 4 Mathematics exams this term and her marks were: $6.5$; $8.2$; $7.4$ and $9.1$.
- a) What is her current mean mark?
- b) If there is still one final exam left, what mark must Aisha get so that her final mean is exactly $8.0$?
A group of 10 friends has a mean age of exactly $14$ years. When the older brother of one of them joins the group, the mean increases to $15$ years. a) Calculate the age…
A group of 10 friends has a mean age of exactly $14$ years. When the older brother of one of them joins the group, the mean increases to $15$ years.
- a) Calculate the age of the older brother.
- b) If a girl aged $10$ then joins, what will be the new mean of the group of 12 people?
- c) Is it possible that, after the girl joins, the median of the group remains $14$? Give an example of the ages of the 10 original friends that satisfies this condition.
Mean from frequency tables and intervals
When the data are in a table, each value counts according to its frequency: $\bar{x} = \dfrac{\sum x_i\, n_i}{N}$. Only the mean of grouped data (using class marks) is Higher Level (see the box below).
The following table shows the number of books read by the students in a 2nd Year class during the first term: | Books read ($x_i$) | Number of students ($f_i$) | | :---…
The following table shows the number of books read by the students in a 2nd Year class during the first term:
| Books read ($x_i$) | Number of students ($f_i$) |
|---|---|
| $0$ | $4$ |
| $1$ | $10$ |
| $2$ | $7$ |
| $3$ | $4$ |
Calculate the arithmetic mean of the number of books read per student.
In a basketball training session, the number of three-pointers scored by each player in 25 attempts was recorded. The results are shown in the following graph: From the…
In a basketball training session, the number of three-pointers scored by each player in 25 attempts was recorded. The results are shown in the following graph:
From the bar chart:
- a) Draw up the absolute frequency table.
- b) Calculate the mean, median, and mode of the three-pointers scored.
You can do it on paper and send a photo with your answer.
A survey was conducted on the daily time (in minutes) that teenagers at a high school spend using social media. The data have been grouped in the following table: | Time…
A survey was conducted on the daily time (in minutes) that teenagers at a high school spend using social media. The data have been grouped in the following table:
| Time (min) | $[0, 30)$ | $[30, 60)$ | $[60, 90)$ | $[90, 120)$ |
|---|---|---|---|---|
| No. of Students ($n_i$) | $12$ | $25$ | $x$ | $7$ |
Knowing that the mean usage time is $57$ minutes:
- a) Calculate the value of $x$ (absolute frequency of the interval $[60, 90)$).
- b) Determine the mode (modal interval) and the median (median interval) of the distribution.
- c) What percentage of students spend $90$ minutes or more on social media?
When we group data, we lose information
Using class marks gives an estimate, not necessarily the exact mean. The specific values inside each interval are no longer known.
The range: the simplest measure of dispersion
The two phone models have the same mean battery life (15 hours), but model A always lasts about the same: range $= 16 - 14 = 2$. Model B is very irregular: range $= 20 - 10 = 10$. For reliability, a small range is a good sign.
The maximum temperatures recorded during a week in April in Cork were: $22$, $25$, $19$, $24$, $27$, $26$ and $21$ °C. Calculate the range of the temperatures for that…
The maximum temperatures recorded during a week in April in Cork were: $22$, $25$, $19$, $24$, $27$, $26$ and $21$ °C. Calculate the range of the temperatures for that week.
In two 2nd Year classes, the time taken by the students to get to school was recorded. In 2A, the times range from $5$ to $40$ minutes, and in 2B, they range from $15$…
In two 2nd Year classes, the time taken by the students to get to school was recorded. In 2A, the times range from $5$ to $40$ minutes, and in 2B, they range from $15$ to $25$ minutes.
a) Calculate the range of the times for each class.
b) In which class are the times most similar to each other? Why?
State whether these statements are true or false, giving reasons and providing examples where appropriate: a) If two data sets have the same mean, they have the same…
State whether these statements are true or false, giving reasons and providing examples where appropriate:
a) If two data sets have the same mean, they have the same dispersion.
b) The standard deviation cannot be negative.
c) If all the values in a data set are equal, its standard deviation is $0$.
d) If we add a value equal to the mean to a data set, the range does not change.
Dispersion beyond the range
The range only looks at the two extremes. The mean deviation takes all the data into account: how far, on average, each value is from the mean.
Box plots and quartiles
The median splits ordered data in two halves. The lower quartile $Q_1$ is the median of the lower half and the upper quartile $Q_3$ is the median of the upper half. The interquartile range is $IQR = Q_3 - Q_1$: a smaller IQR means more consistent data. A box plot shows the minimum, $Q_1$, median, $Q_3$ and maximum.
Variance and standard deviation
To get the variance, find the mean of the squares of the deviations. The standard deviation is its square root.
With a frequency table, weight each squared deviation by its frequency: variance $= \dfrac{\sum n_i (x_i-\bar{x})^2}{N}$.
The number of books read this term by the $20$ students in a class is given by this table: | Books read | $0$ | $1$ | $2$ | $3$ | |---|---|---|---|---| | No. of students…
The number of books read this term by the $20$ students in a class is given by this table:
| Books read | $0$ | $1$ | $2$ | $3$ |
|---|---|---|---|---|
| No. of students | $4$ | $8$ | $5$ | $3$ |
a) Calculate the mean number of books read.
b) Calculate the variance and standard deviation, rounding to two decimal places.
Investigation and spreadsheet: from data to decision
We want to study the minutes spent reading the previous day in two classes of 20 students. We pick 4 people at random from each class and ask the same question, for the same period and in the same unit. We record the answers without names and note any absences: we do not replace them with zero. Choosing at random reduces bias, but a sample of eight cannot guarantee a perfect representation of all 40 people; a single day does not describe all reading habits either.
This activity goes beyond the Junior Cycle specification where it uses the mean deviation (row 5 and column B), which is an extension. Practise with these fictitious data: 20, 30, 30, 40, 40, 40, 50, 70 minutes.
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In A1 write
Minutesand in A2:A9, the data. In C1 writeMeasureand in D1,Result. -
Complete C2:D4 with the first three rows of the table below.
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In B1 write
Distance from the mean; in B2,=ABS(A2-$D$2). Copy down to B9: the column should show 20, 10, 10, 0, 0, 0, 10, 30. -
Complete C5:D6. To build a frequency table, write the values 20, 30, 40, 50, 60, 70 in F2:F7, and in G2,
=COUNTIF($A$2:$A$9,F2). Copy down to G7: the frequencies are 1, 2, 3, 1, 0, 1. -
Add headings in F1:G1 and create a bar chart with separate bars, with "Minutes recorded" and "Students" on the axes. State in the title that there are eight responses and start the frequency axis at zero.
| Row | Measure in C | Formula in D | Result |
|---|---|---|---|
| 2 | Mean | =AVERAGE(A2:A9) |
40 min |
| 3 | Median | =MEDIAN(A2:A9) |
40 min |
| 4 | Mode | =MODE(A2:A9) |
40 min |
| 5 | Mean deviation | =AVERAGE(B2:B9) |
10 min |
| 6 | Range | =MAX(A2:A9)-MIN(A2:A9) |
50 min |
Use the equivalent functions if your application uses different names or separators. Here you are calculating with individual data: they are not estimates using class marks.
Reasoned conclusion: in the first data set, 40 minutes is the centre according to all three measures, but not everyone reads the same amount. We cannot conclude that every student in the school reads 40 minutes, or that one class reads more than another, without comparing their data separately. To guide a school activity, we would enlarge the sample and collect data over several days; we would not use these fictitious data as real evidence.
Cooperating, arguing and managing effort
In teams of three, rotate the roles of data collection, checking and explaining. Each person proposes one conclusion and one limitation; justify the final choice with data. If there is disagreement, try both proposals and listen before replying. Provide readable tables and enough time for the whole team to take part, without assuming who is good at maths because of gender or speed.
If a formula fails, name the difficulty: do you not understand the data, the range or the operation? Simplify the example, check it by hand and change one thing at a time. When you finish, explain which strategy helped you and which mistake taught you something new.
The marks in Salma's last $5$ English exams are: $6$, $7$, $8$, $5$ and $9$. a) Calculate the mean mark. b) Calculate the deviation of each mark from the mean (in…
The marks in Salma's last $5$ English exams are: $6$, $7$, $8$, $5$ and $9$.
a) Calculate the mean mark.
b) Calculate the deviation of each mark from the mean (in absolute value) and the mean deviation.
A health app has recorded the daily steps (in thousands) of a teenager over the last two weeks: $8, 10, 12, 8, 9, 11, 15, 8, 10, 12, 14, 8, 9, 10$ Represent these data…
A health app has recorded the daily steps (in thousands) of a teenager over the last two weeks:
$8, 10, 12, 8, 9, 11, 15, 8, 10, 12, 14, 8, 9, 10$
Represent these data in a frequency table and calculate:
- a) The range of the distribution.
- b) The mean deviation ($DM$) from the arithmetic mean.
Below is a box plot representing the marks obtained by the students of a class in a resit exam: Observe the graph and identify the following measures of position: a)…
Below is a box plot representing the marks obtained by the students of a class in a resit exam:
Observe the graph and identify the following measures of position:
- a) Minimum value and maximum value.
- b) First quartile ($Q_1$), median ($Me$), and third quartile ($Q_3$).
- c) What percentage of students obtained a mark higher than $8$?
The goals scored by a futsal team in their last $5$ matches were: $2$, $5$, $1$, $4$, and $3$. a) Calculate the mean number of goals per match. b) Calculate the variance…
The goals scored by a futsal team in their last $5$ matches were: $2$, $5$, $1$, $4$, and $3$.
a) Calculate the mean number of goals per match.
b) Calculate the variance and standard deviation, rounding to two decimal places.
Cian compares two transport apps to go to school. On his last $8$ trips, the waiting times in minutes were: App A: $4$, $6$, $5$, $7$, $5$, $6$, $4$, $7$ App B: $2$,…
Cian compares two transport apps to go to school. On his last $8$ trips, the waiting times in minutes were:
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App A: $4$, $6$, $5$, $7$, $5$, $6$, $4$, $7$
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App B: $2$, $9$, $3$, $10$, $4$, $8$, $3$, $5$
a) Calculate the mean waiting time for each app.
b) Calculate the standard deviation of each, rounding to two decimal places.
c) If Cian has exactly $10$ minutes available each morning, which app would you recommend to him? Reason your answer using the mean and dispersion.
Go back to the start: the two phones with the same mean are told apart by the range (or standard deviation): the one with the smaller spread is more reliable.