TASK – Measures of Central Tendency

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```					Work in pairs on the following questions and be prepared to take part in a discussion with another pair.

(1) What does the word “average” mean to you? Why might it be useful to know the average value? When is it not
helpful to know the average value?
I’m average. I’m the
I’m average.             30% of us are above
We are average.              I’m the middle           average.
We are the mode.             one. I’m the
Everyone above us            median.
is above average.

105cm           105cm         112cm           129cm           137cm         178cm        193cm

(2) The mean, median and mode are all averages (measures of central tendency). They all have different values for
the people shown above. Discuss how you think these averages are defined (how were they calculated or found?)

(3) Which, if any, is the best type of average to use to describe the height of this group of people? Explain your choice.

(4) Think of a situation in which the mode is the most useful type of average to use. Think of another situation in which
it is the least useful type of average to use.

(5) Think of one major disadvantage of the mean as an average. Think of a situation where the values of the mean and
the median are very likely to be close.

(6) Why do you think the mean is used as an average more often than the median or mode?

(7) Which do you think will be larger and why, the mean or the median?

See Ms. Makunja when you have shared as a pair.
From a list of raw data:
Follow these steps to find the mean height and the median height (in cm) of the seven people on page 39:
105, 105, 112, 129, 137, 178, 193
STAT EDIT 1:Edit ENTER Highlight heading L1 CLEAR ENTER
then carefully type the seven values into list L1 (note: they don’t have to be arranged in any particular order)
STAT CALC 1:1-Var Stats ENTER 2nd 1 ENTER
To see x  137 (the mean is 137 cm) and n=7 (you entered 7 values) and then scroll down to see Med = 129 (the
median is 129 cm)

From a frequency table:
Here are the scores achieved by 20 students in a test:

3, 8, 9, 1, 4, 2, 7, 6, 5, 9, 10, 3, 4, 6, 2, 8, 7, 6, 3, 7

Here is the data organised into a frequency table:

Score out of ten
1      2        3     4       5      6       7      8      9      10
(x)
Frequency (f)           1      2        3     2       1      3       3      2      2       1

Follow these steps to find the mean the median scores of the group of students:

STAT EDIT 1:Edit ENTER Highlight heading L1 CLEAR ENTER
then type the integers from 1 to 10 into list L1
Then type the frequencies shown in the table (1, 2, 3, 2, and so on) into list L2
STAT CALC 1:1-Var Stats ENTER 2nd 1 , (comma key is above the 7) 2nd 2 ENTER
To see x  5.5 (the mean is 5.5 cm) and n=20 (you entered 20 values) and then scroll down to see Med = 6 (the
median is 6 cm)

Question
Which of the measures of central tendency is more affected by the presence of an outlier? Give an example
of data sets where outliers make a big difference.

Group Activity
Work in pairs, discussing the following 4 questions. You have 20 minutes to do this and then you should be ready to
participate in a whole class discussion.

(1)     Here again are the scores achieved by 20 students in a test:

Score out of ten (x)     1      2        3     4       5      6       7      8       9     10
Frequency (f)         1      2        3     2       1      3       3      2       2      1

We’ve already learnt how to find the mean score using the TI83 but how could the mean be obtained without the
calculator?

Explain your method and why it works. For help with this you may refer to your text, page 432 and read example 6.
(2) Look again at the test scores of the 20 students above. You should have found that the mean score is 5.5 so we
would expect the median to be similar.

To find the median we usually list the values in ascending order and then locate the one in the middle position. We
could do that with this data. We know that the first value is 1, the next two are 2 and so on:

1, 2, 2, 3, 3, 3, 4, 4, 5, 6, 6, 6, 7, 7, 7, 8, 8, 9, 9, 10

Since there are 20 values (an even number) there will be a middle pair (the 10 th and 11th values). Counting along the
list we find that the 10th value is 6 and the 11th value is also 6. So the median is in the middle of 6 and 6, so the median
is 6.

We didn’t have to write out the whole list to locate the 10th and 11th values. We could have obtained this information
from the frequency table. How?

For help with this section you may refer to you text, page 432, and read examples 8 and 9.

(3) Finding the mean of grouped data

The following data summarises the distances travelled by a fleet of 190 buses before experiencing a major breakdown.

Distance d (in
40<d 60        60<d80     80<d100          100<d120   120<d140      140<d220
thousands of miles)
Midpoint (x)                  50              70            90               110        130             180
Frequency (f)                 32              25            34               46         33              20

 Is distance a discrete or a continuous variable?

   Look at the second class, 60 < d  80. What exactly does this mean? What are the upper and lower class
boundaries?

   Show how the midpoints have been calculated

   It is not possible to say exactly how far each of the 190 buses travelled before experiencing a major breakdown
using this data. Why not?

   Why do you think we are calculating midpoints of classes?

   Can you see how we are going to find the mean distance travelled before a breakdown? Why is this an estimate of
the mean distance and not the actual mean distance?

For help with this section you may refer to you text, page 440, and read example 10.

(4) Use your knowledge gained from above to find estimation for:
a. the modal class
b. the median
c. the mean age of the bus drivers.
How could you use your TI83 to obtain this estimate for the mean from the grouped frequency table shown?

Ages of bus drivers data, to the nearest year:

Age (years)      21-25   26-30   31-35    36-40   41-45     46-50   51-55
Frequency (f)    11      14      32       27      29        17      7
Age (years)    Frequency (f)   Midpoint (x)     fx        Cumulative (f)
21-25             11             23                         (f)
26-30             14             28
31-35             32             33
36-40             27             38
41-45             29             43
46-50             17             48
51-55              7             53

Using correct IB notation write a formula for calculating the mean of a set of data from a frequency table.

See Ms. Makunja when you have generalized your formula.

Practice:

Exercise 18B.1 Page 429. Q. 4, 5, 7, 10, 13

Exercise 18B.2 Q. 14, 20, 22

Exercise 18B.3 Q 2, 7 (by hand and check with technology)

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