Statistical conceptsWhat students (and teachers) don’t

know

Centre of a distribution What do we mean by “centre”?

Score playing first game of SKUNK

Centre of a distribution The centre is the one best number to

describe the position of the whole group.

Score playing first game of SKUNK

Centre of a distribution Mean score 50.1, median 55 Which is better? Mean or median?

Score playing first game of SKUNK

Centre of a distribution The mean is a more efficient measure

than the median. The sample mean tends to be a better

estimator of the population mean than the sample median is of the population median.

This means that confidence intervals for the mean tend to be narrower than for the median.

Mean or median?

Comparing length of Lake Taupo trout in 1995 and 1998 – can you make a call?

Spread of a distribution What do we mean by “spread”?

Score playing SKUNK with a strategy

Spread of a distribution Spread describes how far the values in

the group are from the centre, how variable they are.

Score playing SKUNK with a strategy

Range is not a useful measure of spread because it is determined only by extreme values.

Students should not use range in any NCEA standard (except the numeracy unit standards).

IQR measures the spread of the whole distribution

IQR is calculated using the width of the middle 50% but it is a measure of the variability of the whole group (just as SD measures the variability of the whole group).

Score playing SKUNK at first and then with a strategy

Shift and overlap are comparisons of centre

Shift answers the question “Which is bigger?”

Overlap answers the question “How much bigger, relative to the spread?”

Score playing SKUNK at first and then with a strategy

Beyond centre and spreadStatistical error is the difference between the sample statistic and the (unknown) population

parameter.

What is sampling error?

What is sampling error?It depends where you ask. It is defined differently in different countries.

In NZ (from Statistics NZ): Sampling error arises due to the

variability that occurs by chance because a random sample, rather than an entire population, is surveyed.

Non-sampling error is all error that is not sampling error.

Non-sampling errorNon-sampling error is all error that is not sampling error.Non-sampling error includes bias due to: A sampling frame which does not

represent the population Sampling method The sampling process and anything else except sampling

variability and choice of sample size.

Sample size There is no statistical basis for insisting on a

sample size of 30. A sample doesn’t have to be very big to give a

rough estimate of the centre of the population. A comment that a bigger sample size would

give a better estimate of the population centre would have to be justified by explaining why it would be important to have a better estimate in that context.

Sample size needs to be fairly large (over 200) to get a reasonable estimate of the population distribution.

Maths teachers teaching literacy

How do we teach students to cope with unfamiliar contexts in exams?

How do we teach students to cope with unfamiliar contexts in exams?

One approach is to start with problems they can do in familiar contexts.1. Solve the familiar problem, then replace the

familiar context with an unfamiliar one, one word at a time.

2. Give them a problem in a familiar context beside an identical problem in an unfamiliar context.

3. Give them the familiar problem followed by the unfamiliar problem.

4. Give them a mix of problems in familiar and unfamiliar contexts.

L2 probability examA pilot study investigated if people showed some symptoms of arthritis.The results were summarised in the table shown below.

gender No symptoms of arthritis shown

Some symptoms of arthritis shown

Total

male 167 33 200female 405 195 600total 572 228 800

Start with a context they know.A student asked if people were sleepy.The results are in the table shown below.

gender Not sleepy

Sleepy Total

male 167 33 200female 405 195 600total 572 228 800

A pilot study asked if people were sleepy.The results are in the table shown below.

gender Not sleepy

Sleepy Total

male 167 33 200female 405 195 600total 572 228 800

A pilot study investigated if people were sleepy.The results are in the table shown below.

gender Not sleepy

Sleepy Total

male 167 33 200female 405 195 600total 572 228 800

A pilot study investigated if people showed some symptoms of arthritis.The results are in the table shown below.

gender Not sleepy

Sleepy Total

male 167 33 200female 405 195 600total 572 228 800

A pilot study investigated if people showed some symptoms of arthritis.The results were summarised in the table shown below.

gender No symptoms of arthritis shown

Some symptoms of arthritis shown

Total

male 167 33 200female 405 195 600total 572 228 800

Side by sideA student asked if people were sleepy.The results are in the table shown below.

A pilot study investigated if people showed some symptoms of arthritis.The results were summarised in the table shown below.

familiarA student asked if people were sleepy.The results are in the table shown below.

gender Not sleepy

Sleepy Total

male 167 33 200female 405 195 600total 572 228 800

followed by unfamiliarA pilot study investigated if people showed some symptoms of arthritis.The results were summarised in the table shown below.

gender No symptoms of arthritis shown

Some symptoms of arthritis shown

Total

male 167 33 200female 405 195 600total 572 228 800

Two-way Venn diagrams are not good problem solving tools.

Students who use two-way tables are much more successful at solvingProbability problems than students who use Venn diagrams.

the end