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Directions: Journal of Educational Studies Vol. 25 Nos 1 &2 June & Dec. 2003
An Exploration of High School Students’ Understanding of Sample
Size and Sampling Variability: Implications for Research
Sashi Sharma
Concerns about students’ difficulties in statistical reasoning led to a study
which explored Form Five (14 to 16-year-olds) students’ ideas in this area.
The study focussed on sampling, probability, descriptive statistics and
graphical representations. This paper presents and discusses the ways in
which these students made sense of sampling constructs obtained from the
individual interviews. The findings revealed that many of the students used
strategies based on prior experiences and intuitive strategies. While they
showed competence with sample size, they were less competent on the
sampling variability task. This could be due to instructional neglect of this
concept or linguistic problems. The paper concludes by suggesting some
implications for further research.

Introduction
In recent years statistics has gained increased attention in our society.
Any newspaper or magazine is likely to contain statistical information.
Decisions concerning business, industry, employment, sports, health, law
and opinion polling are made, using understanding of statistical information
(Wallman 1993). Paralleling these trends, there has been a movement in many
countries to include statistics at every level in the mathematics curricula. In
western countries such as Australia (Australian Education Council 1991),
New Zealand (Ministry of Education 1992), and the United Kingdom (Holmes
1994) these developments are reflected in official documents and in materials
produced for teachers. In line with these moves, Fiji has also produced a
new mathematics prescription at the primary level that gives more emphasis
to statistics at this level (Fiji Ministry of Education 1994). The use of relevant
contexts and students’ experiences and understandings is recommended for
enhancing the students’ understanding of statistics (Watson 2000; Watson
and Callingham 2003). Clearly the emphasis in these documents is on
producing intelligent citizens who can reason with statistical ideas and make
sense of statistical information. Research shows that many students find
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Directions: Journal of Educational Studies Vol. 25 Nos 1 &2 June & Dec. 2003
statistics difficult to learn and understand in both formal and everyday
contexts and that learning and understanding may be influenced by ideas
and intuitions developed in early years (Gal and Garfield 1997). Most of the
research in statistics has been done with primary school children or with
tertiary level students, resulting in a gap in our knowledge about students’
conceptions of statistics at the secondary level. Additionally, since most of
this research has been done in just a very few western countries, it needs to
be determined how culture influences conceptions of statistics; whether biases
and misconceptions of statistics are artifacts of western culture or whether
they vary across cultures.
Concerns about the importance of statistics in everyday life and in
schools, the lack of research in this area and the difficulties students have in
statistical reasoning, determined the focus of my study. Overall, the study
was designed to investigate what ideas a group of Form Five students have
about statistics, and how they construct these. Since the study is qualitative
in nature, the sample was unavoidably small. By adopting a qualitative
approach, I intended to gather rich, detailed and comprehensive data that
enabled me achieve a better understanding of student thinking. The paper
will not make any claims to universality or permanence. Rather, its interest
lies in the specific, idiosyncratic context of this group of high school students.
The study could be viewed as window or lens through which other people
will be able to understand how some students construct meanings for
statistical concepts. Prior to discussing the details of my own research, I will
briefly discuss some existing literature on sampling and variation.
Research on sample size and sampling variation
Gal and Garfield (1997) express the need to study samples instead of
populations and to infer from samples to populations. Additionally, they
suggest that one goal for instruction in statistics is to help students understand
the existence of variation. Metz (1997) and Moore (1990) highlight the key
role that variation plays in students’ understanding of chance. Moore writes
that it is important for students to understand the idea that “chance variation,
rather than deterministic causation, explains many aspects of the world”
(Moore 1990:99). The New Zealand Ministry of Education (2004) states
that since the idea of probability as long-run relative frequency needs to be
addressed with students, variation can no longer be avoided.
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Directions: Journal of Educational Studies Vol. 25 Nos 1 &2 June & Dec. 2003
Although it has been argued that sampling and variation play a
fundamental role in students’ understanding and application of statistics and
chance, little research attention has been given to these concepts (Green
1993; Shaughnessy 1997; Shaughnessy, Watson, Moritz and Reading 1999;
Torok and Watson 2000). Shaughnessy (1997) suggests some possible reasons
for the lack of attention to research on variability concepts. The neglect of
variation is noted in the National Assessment of Educational Progress
(Zawojewski and Heckman 1997) studies which tested student achievement
in grades 4, 8, and 12. The statistics assessment test items addressed concepts
of descriptive statistics, mainly median and mean, as measures of central
tendency, and range as a measure of variability. However, it was a low level
computational task.
To illustrate the undue confidence that people put in the reliability of
small samples, take Tversky and Kahneman’s (1974) problem given to college
students:
Assume that the chance of having a boy or girl baby is the same. Over
the course of a year, in which type of hospital would you expect there to be
more days on which at least 60% of the babies born were boys?
(a)
In a large hospital
(b)
In a small hospital
(c)
It makes no difference
Most subjects in Tversky and Kahneman’s study (1974) judged the
probability of obtaining more than 60% boys to be the same in the
small and in the large hospital. However, the sampling theory entails
that the expected number of days on which more than 60% of the babies
are boys is much more likely to occur in a small hospital because a
large sample is less likely to stray from 50%. According to Tversky
and Kahneman (1974) the representativeness heuristic underlies this
misconception. People who rely on the representative heuristic tend to
estimate the likelihood of events by neglecting the sample size or by
placing undue confidence in the reliability of small samples.
Shaughnessy et al. (1999) surveyed 324 students in grades 4-6, 9 and 12
in Australia and the United States using a variation of an item on the National
Assessment of Educational Progress (Shaughnessy and Zawojewski 1999).
Three different versions of the task were presented in a Before and in a
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Directions: Journal of Educational Studies Vol. 25 Nos 1 &2 June & Dec. 2003
Before and After setting. In the latter setting, students did the task both before
and after carrying out a simulation of the task. Responses were categorised
according to their centre and spreads. While there was a steady growth across
grades on the centre criteria, there was no clear corresponding improvement
on the spread criteria. There was considerable improvement on the task among
the students who repeated it after the simulation. The researchers conjectured
that the lack of clear growth on spreads and variability and the inability of
many students to integrate the two concepts (centres and variation) on the
task may be due to instructional neglect of variability concepts.
To investigate the influence of the concept of sample variability on
students’ thinking as opposed to sample representativeness, Rubin, Bruce
and Tenney (1991) asked 12 high school students to evaluate two different
ways of dividing up 400 runners: 200 fast and 200 slow into blue and red
teams. One was the running ability of each runner and the other was to assign
runners to each team randomly (by choosing names out of a hat and assigning
alternate runners to each team). The students were asked to provide an
assessment of the fairness of the hat method: how likely it was to produce
teams that were balanced in terms of fast and slow runners. Many students
reported that unequal teams were possible with the hat method: teams with
150 fast runners and 50 slow ones were possible outcomes.
It must be reiterated that the research discussed above has been done in
a very few western countries. It would be interesting to determine how
culture influences conceptions of sampling and variability: whether intuitive
strategies and preconceptions such as representativeness are artifacts of
western culture or whether they vary across cultures. Metz (1997) suggests
that to adequately understand students’ cognitive constructions and beliefs,
we need to consider the culture in which students participate. Such
information may help teachers to plan learning activities and students to
overcome their difficulties. In the current interview-based study, both
sampling and variability questions were used to determine specific student
conceptions and the factors that contribute to these constructs.
Overview of the study
The secondary school selected for the research was a typical high school
in Fiji. The class consisted of 29 students aged 14 to 16 years of which 19
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Directions: Journal of Educational Studies Vol. 25 Nos 1 &2 June & Dec. 2003
were girls and 10 were boys. Fourteen students were selected from this class
and this group was representative of the larger group in terms of ability and
gender.
To explore the full range of students’ thinking about sampling concepts,
open-ended questions were selected and adapted from those used by other
researchers. The car question (Item 1) was used to assess whether students
understand that it is more important to base decisions on large samples of
data than on single cases or small samples, however compelling they might
seem. The coin problem (Item 2) was used to explore students’ ideas about
sample variability.
Item 1: The car problem
Mr Singh wants to buy a new car, either a Honda or a Toyota. He wants
whichever car will break down the least. First he read in Consumer Reports
that for 400 cars of each type, the Toyota had more break-downs than the
Honda. Then he talked to three friends. Two were Toyota owners, who had
no major break-downs. The other friend used to own a Honda but it had lots
of break-downs, so he sold it. He said he would never buy another Honda.
Which car should Mr Singh buy? Why?
Item 2: The coin problem
Shelly is going to flip a coin 50 times and record the percentage of heads she
gets. Her friend Anita is going to flip a coin 10 times and record the percentage
of heads she gets.
Which person is more likely to get 80% or more heads?
Each student was interviewed individually by myself in a room away from
the rest of the class. During the interview, care was taken to avoid leading
the students towards any particular viewpoint, so responses to questions were
accepted as they were given and probing questions were asked simply to
ascertain the reasons for what the student thought. The interviews were
recorded for analysis. Each interview lasted between 40 to 50 minutes. Paper,
a pencil and a calculator were provided for the student if he or she needed it.
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Directions: Journal of Educational Studies Vol. 25 Nos 1 &2 June & Dec. 2003
Analysis of data
Analysis of the interviews indicated that the students used a variety of
intuitive strategies and prior knowledge to explain their thinking. I created a
simple four stage-based model that could be helpful for describing research
results relating to students’ statistical conceptions, planning instruction in
statistics and dissemination of findings to mathematics educators. The four
categories in the model are: non response, non-statistical, partial-statistical
and statistical. They are described in Table 1.
Table 1: Characteristics of the four categories of responses
Response Type
Sample size and sampling variation
Non-response
Complete silence, I don’t know, I have
forgotten the rule, I just guessed
Non-statistical responses
Refer to everyday and school experiences
or make inappropriate connections with
other learning areas
Problems with language
Hold the pervasive belief that they can
control outcomes of events
Causality perspective
Belief in luck
Partial-statistical responses
Adapted the rules or applied them
inappropriately.
Refer to representativeness,
unpredictability, equiprobability biases.
Could not explain reasoning
Inconsistent reasoning
Focus on only a small subset of the
information
Statistical responses
Able to justify reasoning by using
frequentist interpretation
Extend sampling rules to unfamiliar
situations
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Directions: Journal of Educational Studies Vol. 25 Nos 1 &2 June & Dec. 2003
The non-statistical responses were based on beliefs and experiences while
the students using the partial-statistical responses applied rules and procedures
inappropriately or referred to intuitive strategies. The term statistical is used
in this paper for the appropriate responses. However, I am aware that such a
term is not an absolute one. Students possess interpretations and
representations which may be situation specific and hence these ideas have
to be considered in their own right. This category has been used mainly to
discuss and present results. Statistical simply means what is usually accepted
in standard mathematics text-books. It would be reasonable to assume another
level (advanced-statistical), equivalent to Shaughnessy’s (1992) pragmatical
statistical level, where students appear to have a very complete view that
incorporates questioning of data but the need for such a category did not
arise in my research and any responses that could have been categorised as
advanced-statistical were simply grouped with the statistical responses.
Results and discussion
This section reports data on students’ understanding of sample size and
sampling variability. In this section the types of responses are summarised
and the ways that the students made their errors are described. Extracts from
typical individual interviews are used for illustrative purposes.
Interview Responses: Item 1
Results of student responses to Item 1 are summarised in Table 2.
Table 2: Response types for task involving sample size (n = 14)
.
Response type
Number of students using it
Non-response
1
Non-statistical
9
Partial-statistical
-
Statistical
4
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Directions: Journal of Educational Studies Vol. 25 Nos 1 &2 June & Dec. 2003
Four students were considered statistical on Item 1. They were able to
apply the law of large numbers and thought that one should believe a consumer
report because it represents a bigger sample.
Non-statistical responses: The nine students who did not use sample size
information on Item 1 based their responses on their beliefs and experiences.
When asked to make predictions, students looked for factors that caused the
behaviour or event under consideration. Rather than attending to sample
size in a consumer report, three students in this study said that Mr Singh
could buy either car because the life of a car depends on how one keeps it.
They did not apply the idea of representativeness in this instance where it is
really appropriate to do so. For example, student 29 explained:
He should buy any of the cars Honda or Toyota; it depends on him how
he keeps and uses the car ... Ah ... Because it depends on the person, how he
follows instructions then uses it. My father used to own a car and he kept it
for ten years. He sold it but it is still going and it hasn’t had any major
breakdowns.

Two students thought that Mr Singh should buy a Toyota. They
generalised inappropriately from experiences with small samples. For
instance, student 17 responded:
Because his friends have had experience with their cars. They are saying
that they had no major break downs. The other friend had Honda and he had
many break downs.

The other four students drew upon their everyday experiences with
consumer reports. Two of these students thought that Mr Singh should take
advice from a consumer report because they were the right people to consult.
The remaining students felt that Mr Singh should not take advice from the
consumer reports because consumer people often give misleading
information. For instance, student 9 explained:
Maybe they want to sell Toyota cars first. We read in the Fiji Times that
people want to sell things; they just advertise that they are good. When people
buy it, is not like that.

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Directions: Journal of Educational Studies Vol. 25 Nos 1 &2 June & Dec. 2003
Interview Responses: Item 2
Results of student responses to Item 2 are summarised in Table 3.
Table 3: Response types for task involving sample variability (n = 14)
Response type
Number of students using it
Non-response
1
Non-statistical
6
Partial-statistical
7
Statistical

One student said that Anita and Shelley were equally likely to get
80% or more heads. However, when asked to explain, she said that she had
just guessed the answer. As Table 2 shows, no student managed to respond to
this problem in a statistical manner. The responses of the other 13 students
were roughly evenly divided between non and partial-statistical.
Non-statistical responses: From a statistical point of view, more than
80% heads is more likely to occur in the small sample because the large
sample is less likely to stray from 50%. However, the results of this study
indicate that six students based their reasoning on their beliefs and
experiences. Two students judged that the probability of obtaining more than
80% heads was more likely to occur with 50 flips of a coin than with 10
flips. The students did not attend to the effect of sample size on variability
when making estimates of the likelihood of outcomes. Thus, the base rate
data of 80% variability was neglected because it did not have any causal
implications. The explanations given by student 17 are indicative of the
causality perspective:
Yeah, Oh … would be Shelley. Because Shelley’s amount comes to 50;
Anita does it only 10 times. Oh … Shelley because she does more flips. She
got more chance to get 80%.

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Directions: Journal of Educational Studies Vol. 25 Nos 1 &2 June & Dec. 2003
The other four students with responses in this category thought that the
flipping of coins depends on luck or how one tosses the coin. For example,
student 20 explained:
Eh ...as I said before… because when you throw each time the coin
comes head or tail or tail or head. She will throw the coin in one direction so
she will get HHH, when she will change in another direction she will get all
tails.. So it will depend on how fast you throw and how fast the coin swings.

Student 6 offered the following response:
This flipping of coins depends on luck; if a person ... is a lucky person
then he will be able to have heads.
Partial-statistical responses: Of the seven students with partial-statistical
responses on this task, one applied rules inappropriately and five based their
reasoning on intuitions such as equiprobability and unpredictability. The
particular rule applied inappropriately by these students was the percentage
rule. For example, student 5 responded that Shelly is more likely to get 80%
because she gets 40. She simply calculated 80% of 50, getting an answer of
40. May be the student did not understand the question and readily performed
arithmetic operations on the numbers given in the problem.
The item produced very strong unpredictability reasoning. From the
students’ explanations, it is clear that their understanding of variation in
small samples was minimal in this context. Student 25 believed that both
Shelly and Anita were likely to get 80% or more heads because I don’t know
what will come. It can be tail or head.
Student 29 thought likewise:
It can be anyone because she tossed the coin 50 times, she can get more
heads, and even this one too [meaning Anita] she can get less tails too eh;
less heads. Both have 50% chances of getting heads.

Another common strategy used can be classed as equiprobability. Three
students thought that neither of them could get 80% heads. Part of the
explanations provided by the students seems to indicate a view that chance
is naturally equiprobable. Even repeated probing by the interviewer did not
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Directions: Journal of Educational Studies Vol. 25 Nos 1 &2 June & Dec. 2003
induce any statistical thinking. Since there are two outcomes in the sample
space, these students believe that each should occur about a half of the time,
regardless of the sample size. Two students even altered their data in this
problem to align it with their personal preferences. When asked which person
is more likely to get 80% or more heads, student 2 responded that both of
them would get the same:
Interviewer: Can you explain your answer?
Student 2:

Because a coin has two sides only heads or tails.
One tosses it 10 times or 100 times, it makes no difference.

Interviewer: So you think that both will get 80% or more heads?
Student 2:

80% or more heads, no [laughs]. Not 80%. They will be
getting 50% or more heads.

Interviewer: Can you say why?
Student 2:

Because the chances of heads and tails is half. So it will
be 50%.

One student gave the correct answer with partially correct reasoning:
Anita, because she does it fewer times.
It must be acknowledged that the limited use of statistical reasoning on
this question may be a consequence of a lack of emphasis on variation in the
classroom and curricular materials. Furthermore, students are not used to
explaining their thinking or perhaps they had difficulty in explaining their
reasoning in detail due to language difficulties. It takes considerable self-
confidence to say something like I can’t describe my thinking in words or I
don’t understand the question.
Gal (1998) states that suggesting to students
that a judgment is called for, rather than a precise mathematical response,
will make students think more about data and not look straight away for
some numbers to crunch.
Sampling and variability: a broader context
Intuitive Strategies: According to Tversky and Kahneman (1974) and
Shaughnessy (1997) the representativeness strategy underlies the sample
variability misconception. The results of this study provide evidence that
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students did not rely on the representativeness strategy but based their thinking
on the unpredictability and equiprobability bias. One explanation for this
could be classroom emphasis on classicist probabilities rather than frequentist
approach. Students can reason about the unpredictability of a single event
but fail to conceptualise the patterns that can emerge across a large number
of repetitions of the event (Metz, 1997). In short, they are unable to integrate
uncertainity and pattern aspects into the sampling construct. Another possible
explanation for this could be that the contexts for the tasks were quite different
and the students were different ages with different statistical backgrounds;
hence the strategies employed were different.
Outcomes can be controlled: The results show that quite a number of students
think that outcomes on random generators such as coins or dice can be
controlled by individuals. The general belief is that results depend on how
one throws or handles these different devices. The finding concurs with the
results of studies by Shaughnessy and Zawojewski (1999). Although this
study provides evidence that reliance upon control assumption can result in
biased, non-statistical responses, in some cases this strategy may provide
useful information for other purposes. For example, student 20’s knowledge
of physics may have been reasonable. The students using this approach have
drawn on relevant common sense information. The responses raise further
questions. Is there a weakness in the wording of this task in that it is completely
open-ended and does not focus the students to draw on other relevant
knowledge? Perhaps, including cues such as ‘fair’ (i.e. not loaded) in item 2
would have aided in the interpretation of this question. Are the students aware
of the differences in probabilistic reasoning compared with reasoning in other
curriculum areas? Although we consider the flip of a coin and the throw of a
die as random, deterministic physical laws govern what happens during these
trials. We can imagine throwing a coin in a way that we can predict the
outcome (pushing them smoothly from a height of 1 cm). It depends on the
situation and the context. Even with a ‘fair’ coin, the side that it lands on is
virtually completely determined by a number of factors such as which way
up it started and the degree of spin. If we knew all this then with sufficient
expertise in physics we could write down some equations which are thought
to govern the motion of the coin and use these to work out which way up the
coin should land. It does not make sense to say that the coin has a probability
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Directions: Journal of Educational Studies Vol. 25 Nos 1 &2 June & Dec. 2003
of one-half to be heads because the outcome can be completely determined
by the manner in which it is thrown.
Relevant contexts: In the study described here, background knowledge that
is often invoked to support a student’s mathematical understanding is getting
in the way of efficient problem-solving. Given how statistics is often taught
through examples drawn from ‘real life’ teachers need to exercise care in
ensuring that this intended support apparatus is not counterproductive. This
is particularly important in light of current curricula calls for pervasive use
of contexts (Meyer, Dekker and Querelle 2001; New Zealand Ministry of
Education 1992) and research showing the effects of contexts on students’
ability to solve open-ended tasks (Cooper and Dunne 1997; Sullivan,
Zevenbergen and Mousley 2002). For instance, the study by Cooper and
Dunne showed that realistic problems disadvantaged working class children
since middle class children had greater linguistic facility.
Limitations
It must be acknowledged that the open-ended nature of the tasks and the lack
of guidance given to students regarding what was required of them certainly
influenced how students explained their understanding. The students may
not have been particularly interested in these types of questions as they are
not used to having to describe their reasoning in the classroom. Some students
in this sample clearly had difficulty explaining explicitly about their thinking.
Students who realised that Anita was more likely to get 80% or more heads
had a difficult time explaining their responses. The issues of language use
are particularly important for these students, for whom the language of
instruction is a second language for them, one that is not spoken at home.
Another reason could be that such questions do not appear in external
examinations. Although the study provides some valuable insights into the
kind of thinking that high school students use, the conclusions cannot claim
generality because of the small sample. Additionally, the study was qualitative
in emphasis and the results rely heavily on my skills to collect information
from students. Some implications for future research are implied by the
limitations of this study.
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Implications for Further Research
One direction for further research could be to replicate the present study
and include a larger sample of students from different backgrounds so that
conclusions can be generalised.
Secondly, this small scale investigation into identifying and describing
students’ reasoning has opened up possibilities to do further research at a
macro-level on students’ thinking and to develop more explicit categories
for each level of the framework. Such research would validate the framework
of response levels described in the current study and raise more awareness
of the levels of thinking that need to be considered when planning instruction
and developing students’ statistical thinking.
Another implication relates to meaningful contexts. The interview results
show that personalisation of the context can bring in multiple interpretations
of tasks and possibly different kinds of abstractions. At this point it is not
clear how a student’s understanding of the context contributes to his/her
interpretation of sampling data. Research on what makes this translation
difficult for students is needed.
The picture of students’ thinking in regard to sampling is somehow limited
because students responded to only one item related to sample size and one
item related to variability. There is a need to include more items using similar
contexts in order to explore students’ conceptions of sampling in much more
depth. It would be interesting to explore how changing the context of the
task influences the types of responses exhibited by the students.
Researchers can accurately assess their subjects’ understanding through
individual interviews. The interview results provide evidence that students
often experience difficulty when speaking about tables. However, in the
present investigation I overcame these difficulties by restating a task or
changing the wording. This would have not been possible in a written survey.
Finally, the place of statistics has changed in the revised mathematics
prescription. Statistics appears for the first time at all grade levels (Fiji
Ministry of Education, Women, Culture, Science and Technology 1994). Like
the secondary school students, primary school students are likely to resort to
non-statistical or deterministic explanations. Research efforts at this level
are crucial in order to inform teachers, teacher educators and curriculum
writers.
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