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SL 4.3—Mean, median, mode. Mean of grouped data, standard deviation. Quartiles, IQR - IB Questionbank | RevisionDojo

Practice SL 4.3—Mean, median, mode. Mean of grouped data, standard deviation. Quartiles, IQR with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

SL & HLPaper 1

The heights (in cm) of 50 plants are recorded in the following frequency table:

Height $(\mathrm{cm})$	Frequency
$20 \leq h<30$	10
$30 \leq h<40$	15
$40 \leq h<50$	12
$50 \leq h<60$	8
$60 \leq h<70$	5

Estimate the mean height of the plants.

[3]

Determine the modal class.

[1]

Estimate the probability that a randomly selected plant has a height of at least 50 cm.

[2]

Question 2

SL & HLPaper 1

The scores, $x$ , obtained by $15$ students in a Mathematics quiz are shown below. $32, 45, 28, 39, 41, 35, 25, 48, 30, 42, 38, 36, 40, 33, 44$

Find the median score.

[2]

Find the lower quartile and the upper quartile of the scores.

[4]

Calculate the interquartile range of the scores.

[1]

Question 3

SL & HLPaper 1

A dataset consists of the numbers $x_{1}, x_{2}, \ldots, x_{n}$ , with mean $\bar{x}$ and standard deviation $s$ .

If each number is increased by a constant $c$ , show that the new mean is $\bar{x}+c$ .

[2]

If each number is increased by $c$ , show that the new standard deviation is $s$ .

[2]

If each number is multiplied by a constant $k$ (where $k>0$ ), show that the new variance is $k^{2} s^{2}$ .

[3]

Question 4

SL & HLPaper 1

A dataset consists of $n$ values $x_{1}, x_{2}, \ldots, x_{n}$ , with mean $\bar{x}$ and variance $s^{2}$ . A new dataset is created by applying the transformation $y_{i}=a x_{i}+b$ , where $a>0$ and $b$ are constants.

Show that the mean of the new dataset is $a \bar{x}+b$ .

[2]

Show that the variance of the new dataset is $a^{2} s^{2}$ .

[3]

If $a=2, b=5$ , and the original dataset has $\bar{x}=10$ and $s=4$ , calculate the mean and standard deviation of the new dataset.

[2]

Prove that the interquartile range (IQR) of the new dataset is $a$ times the IQR of the original dataset.

[3]

Question 5

SL & HLPaper 1

Consider four integers $a$ , $b$ , $c$ , and $d$ such that $a<b<c<d$ .

Let $a<b<c<d$ where the maximum is twice the range, and the median is 10. Find the value of $a$ for which the mean is 11.

[4]

Let $a$ and $d$ be the same as your answers to part (a), but $b$ and $c$ have been altered. If $a,b,c,d$ create a geometric sequence, find the median.

[4]

Question 6

SLPaper 1

Consider three integers $a$ , $b$ , $c$ which follow an arithmetic sequence respectively.

Show that $b$ is the mean of $a$ and $c$

[2]

if $2b$ is the range, find $a$

[2]

A student said that this forms a geometric sequence of common ratio 2, is that correct? Explain why or why not.

[2]

Question 7

SL & HLPaper 2

The scores of $10$ students on a short mathematics test are recorded below.

$65, 72, 80, 68, 75, 82, 70, 78, 60, 70$ .

Find the mean and standard deviation of these scores.

[4]

Question 8

SLPaper 2

A nutrition researcher investigates the relationship between the amount of protein (in grams) in a breakfast meal and the time (in minutes) after which a person begins to feel hungry again. The following data were obtained from eight participants.

Protein (g) $x$	10	14	18	21	25	28	32	36
Hunger time (min) $y$	60	75	82	90	110	115	124	130

Using technology, find
(i) the mean and standard deviation of $x$ and $y$ ;
(ii) the value of the Pearson product–moment correlation coefficient $r$ .

[3]

Find the equation of the regression line of $y$ on $x$ in the form $y=a x+b$ .

[2]

A new breakfast bar contains 30 g of protein. Estimate, using your regression model, how long it will take for an average person to feel hungry again.

[1]

The researcher claims that hunger time increases by about 2.5 minutes for each additional gram of protein. Test this claim against your model and comment on whether it is supported.

[2]

Calculate the coefficient of determination, and interpret its meaning in context.

[2]

Question 9

SLPaper 1

Consider the following data sets $X,Y$ such that $X=\{x-4,x,2x,x+9\}$ and $Y=\{y,2y,3y-6\}$

Find the mean of $X$ and $Y$ in terms of $x$ and $y$ respectively.

[2]

Explain what happens to the mean and variance if all the values increase by a constant $k$

[2]

Given the means of $X$ and $Y$ are equal and that $x$ and $y$ are also equal, find the actual values of the means.

[3]

Question 10

SLPaper 1

A grouped frequency distribution of the ages of participants in a survey is given below:

Age Group	Frequency
10-19	5
20-29	12
30-39	18
40-49	10
50-59	5

Calculate the mean age of the participants.

[4]

Determine the median age group.

[2]

Calculate the interquartile range (IQR) of the ages.

[3]

SL 4.3—Mean, median, mode. Mean of grouped data, standard deviation. Quartiles, IQR Questionbank