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Helping students develop statistical reasoning through Seeing Statistics®

Ken W. Li
Department of Computing & Mathematics
Hong Kong Institute of Vocational Education (Tsing Yi)
Students generally focus only on the mechanics of statistical calculations and graphing. They know how to carry out statistical calculations, draw statistical graphs and charts, and use computer software to produce statistical results, but they are unable to express their statistical works as well as interpretation of the statistical findings. To accomplish these tasks, students need to acquire the ability of reasoning with data and statistical ideas. However, the traditional classroom teaching can hardly help them grasp the essentials of statistical concepts. It is believed that more concrete statistical concepts can be vividly illustrated via computer technology. In this paper, the author discusses how computer technology, for example Seeing Statistics(tm), can be employed to elaborate on statistical concepts, promote interactive instruction, and make e-learning more interesting and meaningful.


Students generally know how to carry out statistical calculations, draw statistical graphs and charts, and use computer software to produce statistical results. However, they may not be able to express their statistical works and interpretation of the statistical findings in words. They do not even check whether or not the statistical results produced by computer software are proper in terms of appropriate selection and correct use of statistical tools (Ben-Zvi and Friedlander, 1996). It seems that they only focus on routine or mechanical applications of statistical and computer tools. In reality, they are expected to have acquired statistical reasoning and to be able to communicate statistically (Bradstreet, 1996).

A thorough grasp of statistical concepts is essential for developing statistical reasoning skills (Cobb, 1991; Bradstreet, 1996). Traditional blackboard classroom teaching can hardly explain statistical concepts in words and diagrams for the following two reasons. Usually, class time is tight in each lesson, and drawing diagrams takes much time, not to mention that some diagrams are not so easy to visualize. Some particular statistical experiments are not easily carried out in real world situations. But with the aid of computer technology, students are more easily convinced of statistical concepts through their experimentation with data and interactive computer demonstration (Ben-Zvi and Friedlander, 1996).

Use of computer technology

The World Wide Web (WWW) is an instructional delivery platform for equipping the students with a range of skills required at work as well as in learning. The strengths and merits of instructional delivery via WWW are:- to provide increased flexibility for individual learning; to help teachers create a motivating educational environment for students; to provide channel for significant student-teacher and student-student interactions; to create web-based assessment; to provide immediate positive feedback on student's action; to reinforce the students with correct response; to maintain students' interest in learning; and to keep an up-to-date student record.

To accomplish these goals, WebCT is an appropriate authoring tool for developing web-based learning materials. It requires little technical expertise of the developer to create new or edit existing learning materials. Using WebCT as a learning tool, students need not have much computer literacy to interact with the electronic learning materials. Some of its main features include:- supporting multimedia learning environments; creating searchable and linkable glossary; providing on-line quizzes with feedback; keeping each individual student's record of learning progress; managing database of student performance; and facilitating student-student and student-teacher communications.

The major weaknesses of web-based learning materials are the bandwidth problem and the traffic jam in the information superhighways (Nott et al., 1995). Students waste a great deal of time in downloading data during an on-line learning session, whereas learning materials stored in CD-ROM format can provide instantaneous responses. One of the major weaknesses of CD-ROM educational software is that its version may not be up-to-date unless the users are well informed of the release of the latest version.

Fuller (1997) summarizes how to use computer technology to enhance statistics teaching and learning. He also points out that effective statistics teaching still relies on its instructional design, instructional delivery and student's learning strategy.

Instructional design and delivery

Teachers must have an understanding of how learning materials be arranged using computer technology so as to provide an effective learning environment. They should be aware of the coverage of statistical concepts and techniques in the syllabus. Sequencing subject contents in a sensible order is a necessary condition for successful learning because students should firstly master the basic statistical concepts and knowledge. With these solid foundations, they should be capable of studying further and more difficult subject materials.

According to one of Maehr's principles on students' motivation to learn (1984), the design of computer-based learning materials should be well structured and task oriented. A good approach to learning is through a sequence of clear and concise tasks and demonstrations. Students are expected to go through the material of each topic in a systematic manner that leads them to building up their confidence and foundation in mastering statistical concepts and skills. Eventually, they should be able to achieve some specific learning objectives at the end of each lesson.

The rapid development of computer technology enables teachers to create a motivating educational environment for students. They appear to be more motivated when they are working with computers because of the game like atmosphere, exciting visual displays and so on as generated by multimedia technology. Once motivated, students can learn more and at a better pace (Hess and Tenezakis, 1970; Linskie, 1977). Computer-aided instruction enables teachers to make good use of texts, graphics videos, sounds and animation to teach statistical concepts. It makes statistics learning more interesting and contributes to an easy understanding of statistics.

Discovery learning

Many educational psychologists agree that students must understand the structure, the fundamental ideas, relationships or patterns of a topic in order to learn and remember it. Thus, making learning meaningful and memorable for students is a key issue. In Bruner's discovery learning (1971), teachers pose questions to let students try to find the answers by making intuitive guesses, experimenting with data and so forth. When they discover the solution to a problem through their active involvement in learning, they will probably understand and remember the information more concretely. This makes personal discoveries as the basis for true understanding and the value of inductive reasoning in learning.

Statistics learning can be considerably facilitated by activities that students regard as purposeful and interesting because most teachers agree that purposeful and interesting activity is the most efficient type of teaching and learning. Students are usually active to participate in activities that interest them. These activities can be in any form, for example, exploration of data or data experimentation and so on. The activities should grow out of problems in real-life situations to arouse students' statistical interest and stimulate their statistical thinking. This also leads to superior learning (Strike, 1975).

One effective learning strategy is to allow a student to directly refer to glossary, images, etc. related to their learning queries. The student can choose the sequence of referencing that best suits his/her own interests and abilities. This is the best time to teach concepts as a student makes inquiries (Velleman, 1998). Accessing the required information in this fashion is to employ hypertext technology to establish hyperlinks between referenced materials within the WWW. This enables a student to jump around the hypertext within and outside the same document. Within the hypertext learning environment, a student can explore and interact with knowledge in a non-linear and interactive way using graphics, videos or audio and so on. Non-linear information delivery is more appropriate because human information processing is in non-linear manner (Richards and Barker, 1994).

Moreover, statistics learning involves both its processes and its products (Biehler, 1993). It should not only ask students to learn procedural knowledge but also helps them develop the ability of statistical reasoning. In fact, more emphasis should be placed on the processes rather than the end products in the teaching of statistics because statistics learning by rote does not lead to deeper understanding of statistical concepts but misconception (Bournaud et. al., 1994). Applying Bruner's ideas in statistics classroom, a teacher should not only help students see connections among various concepts but also organize their experiences into meaningful patterns. Thus, besides teaching mean, median and mode-the three ways to measure the central tendency of data, a teacher should give out some data sets having different distributions for students to develop some guidelines for judging which one of the three ways best represents the middle of a data set. Through discovery journey of learning like this, what is learned is to develop the statistical reasoning. This can be easily applied to other problem, as there are no fixed guidelines in problem solving.

Discovered concepts give students deeper understanding and are easier to retain or recall. In practice, discovery learning is so difficult to organize successfully. However, with the aid of computer technology in preparation of computer-based learning materials, discovery learning becomes ideals.

Use of Seeing Statistics®

Seeing Statistics (available at is a web book prepared by McClelland (1999) and has a rich resource for statistics teaching and learning. It aims to teach elementary statistics the way that most students can learn. It has sixteen chapters covering the following topics:- 1) Introduction; 2) Data and Comparison; 3) Seeing Data; 4) Describing the Centre; 5) Describing the Spread; 6) Seeing Data, Again; 7) Probability; 8) Normal Distribution; 9) Inference and Confidence; 10) One-Sample Comparisons; 11) Two-Sample Comparisons; 12) Multi-Group Comparisons; 13) Correlation and Regression; 14) Categorical Data; 15) Nonparametrics and Transformations; and 16) Gallery.

Learning with Seeing Statistics is under each individual student's control. Each student learns at his/her own pace and can repeat any activity as often as needed. Seeing Statistics uses interactive component to explain and visualize statistical concepts and offers students hands-on practice exercises. It has seven core features: Contents, Calculator, Glossary, Search, References, Comments and Site Help. Contents give an overview of the topics in a lesson. Calculator is an electronic device for simple calculations. Glossary provides definitions of statistical terms. Search allows audience to search a particular term throughout the book. References provide a listing of statistics books. Comments enable the author to hear about what the audience's comments and/or suggestions.

Seeing Statistics enables students to discover statistical concepts, explore statistical principles and apply statistical methods and techniques. Learning takes place in student participation in activities. Concepts discovered by students are easier to retain and recall from memory. This can be readily integrated with understanding of other statistical topics.


To help students develop statistical reasoning, it would be better for students to grasp conceptual understanding of statistics (Cobb, 1991; Bradstreet, 1996). Normal Distribution is a very important topic in statistics because many statistical tests and methods are based on the normality assumption. Nevertheless, students do not understand the topic thoroughly. The following examples show how Seeing Statistics can be used to help students learn the materials related to the Normal Distribution.

Students can use the following interactive tool (see Figure 1) to visualize how changes of mean and standard deviation affect the location and the shape of a normal density curve. This tool allows them to choose different means and different standard deviations to gain true understanding through their hands-on learning experience. Students can see the effect of their actions while they are working.

Figure 1

Figure 1: Normal curve with different means and standard deviations

Area under normal density curve

Students have problems of checking the probability values from a Normal Table because different statistics books use different conventions (e.g., Newbold, 1995; Mendenhall et al., 1996; Freund & Walpole, 1987; Milton & Arnold, 1986; etc.). Different Normal tables report the probability value associated with the area under the normal curve differently, i.e., cumulative probability, one-sided probability, two-sided probability or middle score. The following visualisation tool enables students to gain genuine understanding of the probability value under the normal density curve (see Figure 2).

Figure 2

Figure 2: Area under Normal curve

Sampling distribution of mean

The concept of sampling distribution is important in statistical inference but not easy to understand. To facilitate learning, run the following simulation studies for different sample sizes, say 1 and 2 (see Figure 3). The sampling distribution of mean will be normal and the expected value of the mean is an unbiased estimate of the population mean.

Figure 3

Figure 3: Sampling Distribution with n=1 and n=2

Normal approximation to binomial distribution

When n (the sample size) is large and p (the probability for each independent trial) is small, calculating the probability value for a long trial under the binomial distribution becomes tedious. Thus, an alternative is to use the normal approximation to the binomial distribution. However, students are sometimes puzzled about this approximation because binomial distribution is a discrete probability distribution while normal distribution is a continuous one. To convince the students, the following interactive tool can be employed to demonstrate binomial distribution getting closer to normal distribution when n is large and p is small (see Figure 4).

Figure 4

Figure 4: Normal Approximation to Binomial Distribution

Central Limit Theorem

Central Limit Theorem is one of the most important theorems of statistics. It helps statistical users realise that the sampling distribution of the mean is approximately normal if the sample size is large, irrespective of underlying distribution of the sample. It is difficult to teach this in a traditional classroom setting. Fortunately, teachers now can run simulation studies to demonstrate its worthiness (see Figure 5).

Figure 5

Figure 5: Central Limit Theorem


After learning a topic, it is about time to check how well the concepts of normal distribution an individual student has learned. The following exercise poses to students a question to test their understanding. The exercise also serves as additional reinforcement. If students find some concepts vague, they can always review what they have learned by going through the preceding activities (see Figure 6).

Figure 6

Figure 6: Exercise


To demonstrate the power of statistical reasoning, real-life applications to various disciplines such as, Psychology, Engineering and Business, can enhance statistical reasoning. It provides students with opportunities to apply newly learned concepts and methods using real-life data. This process can reinforce their learning while these concepts and methods are still fresh in their mind (see Figure 7).

Figure 7

Figure 7: Application


Students should be able to communicate in statistical terms properly and effectively. They should be at least able to read and understand statistical information; use elementary statistical tools to analyse data; interpret and present their statistical findings. They are expected to acquire statistical reasoning that is a prerequisite in the information age.

Helping students develop statistical reasoning, teachers should pay close attention to the instructional design and delivery of statistics education. Discovery learning is always more effective than learning from lectures. Students go through their own journey of learning to gain a true understanding of statistical concepts. The concepts discovered are easier to recall or retain, and more readily integrated with practical applications of statistics. However, discovery learning is sometimes hard to organise successfully. With the aid of computer technology, more concrete statistical concepts and principles can be illustrated by providing students with hands-on learning and practice experience.


The author would like to thank Prof. S. H. Hou of the Hong Kong Polytechnic University and referees for their valuable comments on an earlier version of the manuscript. He also thanks Prof. G. H. McClelland of Colorado University and Duxbury Press for their permission to use their computer software in the preparation of this manuscript.
Trademarks Notice: Seeing Statistics® is a trademark under licence.


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Author: Ken W. Li, Hong Kong Institute of Vocational Education (Tsing Yi)
Phone: 852-2436-8573 Fax: 852-2435-1406 Email:

Please cite as: Li, K. W. (2001). Helping students develop statistical reasoning through Seeing Statistics. In L. Richardson and J. Lidstone (Eds), Flexible Learning for a Flexible Society, 420-427. Proceedings of ASET-HERDSA 2000 Conference, Toowoomba, Qld, 2-5 July 2000. ASET and HERDSA.

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