Fish figures
This is a level 3 and 4 statistics activity from the Figure It Out series. It is focused on constructing a histogram, using graphs to describe the data, and making comments from a trendline. A PDF of the student activity is included.
About this resource
Figure It Out is a series of 80 books published between 1999 and 2009 to support teaching and learning in New Zealand classrooms.
This resource provides the teachers' notes and answers for one activity from the Figure It Out series. A printable PDF of the student activity can be downloaded from the materials that come with this resource.
Specific learning outcomes:
- Construct a hiistogram.
- Use graphs to describe the data.
- Make comments from a trendline.
Fish figures
Achievement objectives
S3-1: Conduct investigations using the statistical enquiry cycle: gathering, sorting, and displaying multivariate category and wholenumber data and simple time-series data to answer questions; identifying patterns and trends in context, within and between data sets;communicating findings, using data displays.
S4-1: Plan and conduct investigations using the statistical enquiry cycle: determining appropriate variables and data collection methods; gathering, sorting, and displaying multivariate category, measurement, and time-series data to detect patterns, variations, relationships, and trends; comparing distributions visually; communicating findings, using appropriate displays.
Description of mathematics
This diagram shows the areas of statistics involved in this activity.
Investigation |
Literacy |
Probability |
||||
|---|---|---|---|---|---|---|
P |
P |
D |
A |
C |
||
The bottom half of the diagram represents the 5 stages of the PPDAC (Problem, Plan, Data, Analysis, Conclusion) statistics investigation cycle.
Required materials
- a computer spreadsheet/graphing program
- Figure It Out, Levels 3-4, Statistics Revised Edition, "Fish figures", pages 12-13
- a classmate
See Materials that come with this resource to download:
- Fish figures activity (.pdf)
Activity
In these activities, which involve measurement data, the students convert stem-and-leaf graphs into histograms, create a scatter plot, and use trendlines to help them make predictions.
Activity 1
Question 1 is designed to show the relationship between a stem-and-leaf graph and a histogram: rotate the stem-and-leaf graph 90 degrees anticlockwise, adjust the labelling, and there you have it.
This process illustrates an important advantage that a stem-and-leaf graph has over a histogram. In the first, it is possible to see the shape (or distribution) of a data set and still identify individual data values. In the second, the shape of the distribution is just as clear but the individual data values are submerged in the groups (intervals); detail has been lost.
A further advantage of the stem-and-leaf graph is that the data count and the median (and, in larger data sets, the quartiles) can easily be found. The data count cannot be found from a histogram and, for continuous (measurement) data, the median and quartiles can only be estimated.
The students should ensure that the leaves on their stem-and-leaf graph are evenly spaced so that, when it is rotated, it matches the histogram and it is easier to see which intervals have the greatest frequency.
Activity 2
The students need to have access to and an understanding of how to use a computer spreadsheet/ graphing program for this activity (though, for a small data set such as this, a scatter plot can also be drawn by hand reasonably easily).
This may be the first time that your students have come across scatter plots. This type of graph (also known as an XY scatter graph) is designed to show what relationship (correlation), if any, exists between two variables (in this case, the mass and length of fi sh). One of the most useful features of a scatter plot is that each individual data value retains its identity – no data whatsoever is lost in the graphing process. This means that the reader can see both the pattern generated by the data as a whole and trace exactly where every individual piece of data (fi sh, person, book, or whatever) has got to.
So that the students have an idea what to expect, page 13 in the students’ book shows an appropriately formatted graph, with two data values in place. (Needless to say, if the graph is created by computer, all the values appear simultaneously. Note that you can add values and update the computer graph at any stage.)
It is clear from the pattern of dots in the completed graph that as the fish gets longer, the mass increases. If there were no such relationship, the dots would be randomly scattered around the graph instead of clustered very obviously along an imaginary line.
Question 2 involves adding a trendline to the graph and then using it to make two predictions. The predicting provides a check that the students understand the meaning of the graph and the relationship it reveals. (You will probably find Add trendline under Chart on the menu bar of your graphing program.)
Activity 1
1.
The stem-and-leaf graph below shows the lengths of the fish caught.
2.
Descriptions will vary. For example, most of the fi sh weighed between 1 and 3 kg. Two fish weighed less than 1 kg and one weighed 4.4 kg. Most of the fish were between 30 and 50 cm long. The shortest was 20 cm, and the longest was 62 cm.
Activity 2
1.
Your graph should look similar to this (except for the trendline, which has been added here in answer to question 2a):
The pattern shows a definite relationship between length and mass: the longer the length of the fish, the heavier the fish is (within moderate variation); looked at the other way, the heavier the fish, the longer it is.
2.
a–b. See the graph in question 1 for the trendline. The trendline should confirm your answer for question 1.
c. On the basis of the trendline in the graph for question 1, you could expect a fish weighing 3.5 kg to be between 50 and 60 cm in length; a fish that is 48 cm long could be expected to weigh between 2.5 and 3 kg.
"Fish figures" can be used to develop these key competencies:
- thinking
- using language, symbols, and texts
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