Directorate: Curriculum FET
SUBJECT & GRADE
TERM 2
TOPIC
AIMS OF LESSON
RESOURCES
Mathematical Literacy, Grade 11
Week 5
Data handling – Summarising of Data
The week’s lessons will focus on the following aspects of data handling:
• Summarise and compare two sets of data using the measures of central tendency and spread.
• Function and purpose of the measures of central tendency and spread.
• Role and impact of outliers on the measures of central tendency and spread.
• Strengths and limitations of each type of measures of central tendency and spread and when it is most
appropriate to use.
Paper based resources
Digital resources
Via Afrika Learners book:
• Summarising Data: Pages 266 –
269
INTRODUCTION
CONCEPTS AND
SKILLS
Summarising Data:
•
•
https://www.khanacademy.org/math/ap-statistics/summarizingquantitative-data-ap#measuring-center-quantitative
https://www.youtube.com/watch?v=A1mQ9kD-i9I
https://www.youtube.com/watch?v=B1HEzNTGeZ4
•
Study & Master Learners Book:
• Summarising Data: Pages 468 477
• There is no new content in this section from the Gr 10 curriculum. It is expected that the skills and
knowledge gained in Gr 10 must now be used to work with two sets of data instead of just one as in
Grade 10.
• Primary focus in Grade 10 was on learning how to calculate the measures of central tendency and
spread.
• Primary focus in Grade 11 is now using measures of central tendency and spread to compare
different sets or different components within a data set and make deductions.
SUMMARISING DATA
REQUIRED TERMINOLOGY:
• Mean – The sum of all of the data values divided by the number of data values in the set.
Commonly referred to as the average.
• Median – The “middle value” in a sorted (arranged) data set.
•
•
Mode – The number/data value that appears the most in a data set.
Range – The difference between the lowest and the highest value in a data set.
MEASURES OF CENTRAL TENDENCY:
• Mean, Median & Mode
• Indicates a value in the data set that can be seen to be representative and stand for the majority of the
values in the data set.
MEASURES OF SPREAD:
• Range
• Gives an indication how spread out the values in a data set is.
• Only becomes possible to see if values in a set are widely spread out when it becomes possible to
compare the range to another data set.
• The spread of values in a data set can provide important information trends that exist within a data
set.
SUMMARY OF THE MEASURES OF CENTRAL TENDENCY AND SPREAD:
MEASURE
METHOD
MEAN
𝑠𝑢𝑚 𝑜𝑓 𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑑𝑎𝑡𝑎 𝑠𝑒𝑡
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑣𝑎𝑙𝑢𝑒𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑑𝑎𝑡𝑎 𝑠𝑒𝑡
MEDIAN
Middle value in a sorted data set:
• Odd number of values in data set –
median will be one value in the data
set.
• Even number of values in data set –
median will lie between two values
in the middle.
▪
𝒗𝒂𝒍𝒖𝒆 𝟏+𝒗𝒂𝒍𝒖𝒆 𝟐
𝟐
MOST ACCURATE
WHEN
• Not affected when
values are grouped
closely together or
spread widely apart.
• Most accurate when
there are no outliers
present.
• Most accurate when data
is grouped closely
together – middle value
still provides an accurate
indication of the average
value.
• Not affected by outliers
– more effective when
outliers are present
compared to the mean.
INFLUENCED NEGATIVELY
BY
• Outliers affect the mean –
can make the mean
unrealistically low or high.
•
Median can become
unrealistic if the data values
are widely spread apart.
▪ Middle value might
not provide realistic
average for the
values
•
ACTIVITIES/
ASSESSMENT
MODE
Value that occurs most often or
frequently in a data set.
•
RANGE
Highest value – Lowest value
•
Especially useful when
it is important to know
the object that occurs the
most.
Most accurate when
there are no outliers in
the data set.
•
Just because a value appears
most often does not mean it is
representative of the data set.
•
When an outlier is present,
the range can seem to be
unrealistically big or small.
o Creates a false
impression on how
the values are spread
out.
Generally, we calculate all of the measures of central tendency and then compare each measure to the
other values to see which measure provides the most representative indication of the majority of the
values.
CAN YOU DO THE FOLLOWING IN SUMMARISING DATA?
▪ Define the measures of central tendency and spread.
▪ Calculate the measures of central tendency and spread and compare the measures and explain the
differences between the data sets.
▪ Decide which measure of central tendency is most appropriate under a given set of circumstances.
▪ Understand the impact of outliers on the accuracy of the measure of central tendency and/or spread
SUMMARISING DATA:
ACTIVITY 1:
James employs 10 workers by his school tuckshop. Below are the weekly wages of the 10 employees.
R500; R450; R350; R750; R350; R450; R750; R450; R400; R300
1.1:
1.2:
1.3:
Calculate the following:
a)
Mean
b)
Median
c)
Mode
Which indicator(s) provides the most realistic average weekly income of the workers? Explain your
answer.
Which indicator(s) provides an unrealistic average weekly income of the workers? Explain your
answer.
ANSWERS:
500+450+350+750+350+450+750+450+400+300
1.1: a)
𝑀𝑒𝑎𝑛 =
10
4750
1.2:
1.3:
𝑀𝑒𝑎𝑛 = 10
𝑀𝑒𝑎𝑛 = 𝑅475
b)
R300; R350; R350; R400; R450; R450; R450; R500; R750; R750
450+450
𝑀𝑒𝑑𝑖𝑎𝑛 =
2
𝑀𝑒𝑑𝑖𝑎𝑛 = 𝑅450
c)
𝑀𝑜𝑑𝑒 = 𝑅450
The median and the mode gives the most useful indication of the weekly wage. Majority of
the workers earn wages that are similar to these values.
The mean weekly average provides an unrealistic indication of the average wage. Only 3 workers
earn more than R450 per week but the mean average was R475. This is because of the R750 wages
that two workers earn, it positively skews the mean.
ACTIVITY 2:
In preparation for a party to celebrate James’ 40th birthday he asked 20 of his close friends and family
members what their favourite food is. This will give him an idea what he must prepare for his party.
The table below contains the responses of his friends and family members.
2.1:
2.2:
Jaco
Pizza
Kelly
Chicken
Dale
Pizza
Rachel Steak
Kevin
Steak
Jake
Pizza
Simon
Steak
Melissa Pizza
Richard Chicken
Eve
Ribs
Dylan
Chicken Charles Ribs
Mike
Ribs
Elizabeth Steak
Kagiso
Ribs
Peter Steak
Jacques Burgers
Megan
Burgers Margaret Pizza
Victor Steak
Which measure of central tendency should James use to determine which food is most popular?
Use this measure to determine the three most popular food choices.
ANSWERS:
2.1: James should use the modal average as this will show the most popular choice. It is impossible to
use the mean or the median average in this scenario as the mean needs a numerical value and the
median has to be arranged in ascending/descending order and it is not possible in this scenario.
2.2: In order to find the three most popular food choices it is easier to set up a frequency table.
FOOD CHOICE NO OF RESPONSES.
Pizza
5
Steak
6
Chicken
3
Ribs
4
Burgers
2
Now that we have a frequency table we can identify the three most popular food choices as:
1 = Steak (6 responses)
2 = Pizza (5 responses)
3 = Ribs (4 responses)
ACTIVITY 3:
The following graph shows the average annual school fees for government schools per province in South
Africa.
3.1:
3.2:
3.3:
3.4:
Determine the mean annual school fees in South Africa (assume for the purposes of calculations that
all schools have the same amount of schools per province).
Determine the median annual school fees in South Africa. From which province is this value?
Why would it not be possible to determine the modal average for this set of data?
Determine the range of the annual school fees in South Africa.
ANSWERS:
220+90+830+195+95+150+120+370+700
3.1: 𝑀𝑒𝑎𝑛 =
9
2770
𝑀𝑒𝑎𝑛 = 9
𝑀𝑒𝑎𝑛 = 𝑅307,78 𝑝𝑒𝑟 𝑦𝑒𝑎𝑟
3.2:
CONSOLIDATION
VALUES
R90; R95; R120; R150; R195; R220; R370; R700; R830
Median = R195
Province = Kwazulu–Natal
3.3: The mode refers to the value that appears the most, more than any others, in this instance all of the
values appear the same amount of times, therefor there is no mode in the scenario.
3.4: 𝑅𝑎𝑛𝑔𝑒 = 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑣𝑎𝑙𝑢𝑒 − 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 𝑣𝑎𝑙𝑢𝑒
𝑅𝑎𝑛𝑔𝑒 = 𝑅830 − 𝑅90
𝑅𝑎𝑛𝑔𝑒 = 𝑅740
Essence of lesson:
- Summarising Data
i.
Know the definition of the measure of central tendency and spread.
ii.
Know how to determine the measures of central tendency and spread.
iii. Understand the advantages and disadvantages of each measure of central tendency and
spread.
iv.
Be able to compare and decide which is the most appropriate measure to use.
Learners can implement analytical thinking when facing a problem and can make decisions based on
statistical evidence.