# g) Data Handling - Student - school

```Data Handling
Collecting Data
Learning Outcomes
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Understand terms: sample, population, discrete, continuous and
variable
Understand the need for different sampling techniques including
random and stratified sampling and be able to generate random
numbers with a calculator or computer to obtain a sample
Be able to design a questionnaire (taking bias into account)
Understand the need for grouping data and the importance of
class limits and class boundaries when doing so
DH - Collecting Data
Data Handling
Sample:
A sample is a subset of the population. 11A would be a subset of the
following populations → year 11, senior pupils, pupils of St Mary’s
Population:
The total number of individuals or objects being analyzed; this quantity is
user defined. E.g. pupils in a school, people in a town, people in a postal
code.
Discrete:
A discrete variable is often associated with a count, they can only take
certain values – usually whole numbers.
E.g. number of children in a family, number of cars in a street, number of
people in a class.
DH - Collecting Data
Data Handling
Continuous:
A continuous variable is often associated with a measurement, they can
take any value in given range.
E.g. height, weight, time.
Variable:
See discrete & continuous above.
DH - Collecting Data
Data Handling
Random Sampling:
In simple random sampling every member of the population is a given
number. If the population has 100 member , they will each be given a
number between 000 and 999 (inclusive) then 3 digit random numbers are
used to select the sample (ignore repeats)
Stratified Sample:
Often data is collected in sections (strata).
Eg. Number of pupils in a school. In selecting
such a sample data is taken as a proportion of
the total population. Here we should sample
twice as many people in year 10 than in
year 8.
Year
No. of Pupils
8
100
9
50
10
200
11
200
12
150
Total
700
Data Handling
DH - Collecting Data
Stratified Sample:
To obtain as sample of 70 pupils out of the 700, we construct the
following table
Year
No. of
Pupils
8
100
100
9
50
50
10
200
200
/700 = 2/7
100
/700 = 2/7 × 70 = 20
11
200
200
/700 = 2/7
100
/700 = 2/7 × 70 = 20
12
150
150
/700 = 3/14
100
700
Proportion of total No. of pupils to be sampled
/700 = 1/7 × 70 = 10
/700 = 1/7
100
/700 = 1/14
100
/700 = 1/14 × 70 = 5
/700 = 3/14 × 70 = 15
70
DH - Collecting Data
Questionnaires
1. Sample should represent population
2. Sample must be of a reasonable size to represent population
(at least 30) sample mean = population mean
3. Questions should:
i) be as short as possible
ii) use tick boxes
iii) avoid bias
Data Handling
Collecting Data
Learning Outcomes:
At the end of the topic I will be able to
Can
Do
Revise
Further
Understand terms: sample, population, discrete,
continuous and variable


Understand the need for different sampling techniques
including random and stratified sampling and be able to
generate random numbers with a calculator or
computer to obtain a sample


Be able to design a questionnaire (taking bias into
account)




Understand the need for grouping data and the
importance of class limits and class boundaries
Data Handling
Analysing Data
Learning Outcomes
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


Understand that in order to gain a mental picture of a collection
of data it is necessary to obtain a measure of average and range
Be able to determine the mean, median and mode for a set of
raw scores and an ungrouped frequency table
Be able to obtain the median and interquartile range for grouped
data from a cumulative frequency graph
DH - Analysing Data
Measures of
Central Tendency
Mean
Sum of all measures divided by total number of measures.
x

x n
 everyone included
× affected by extremes
Mode
Most popular / most frequent occurrence.
× not everyone included
 not affected by extremes
Median
Arrange data in ascending order; the median is the middle
measure. Position = ½ (n + 1)
× not everyone included
 not affected by extremes
DH - Analysing Data
Measures of
Central Tendency
Examples
Calculate the Mean, Median and Mode for:
a) 3, 4, 5, 6, 6,
b) 2.4, 2.4, 2.5, 2.6
* Normal distribution is where the mean, median and mode are close
eg example b)
DH - Analysing Data
Frequency Distribution
The number of children in 30 families surveyed are surveyed.
The results are given below.
Calculate
a) The mean number
of children per family
b) The median
(No. of children)
x
0
1
2
3
4
5
(No of families)
f
4
5
10
6
3
2
Grouped Frequency
Distribution
DH - Analysing Data
Often data is grouped so that patterns and the shape of the distribution can be
seen. Group sizes can be the same, although there are no applicable rules.
Find the mean of:
Mark
Frequency (f)
30 – 34
7
40 – 49
14
50 – 59
21
60 – 69
9
∑f = 51
Midpoint (x)
fx
Cumulative
Frequency Curves
DH - Analysing Data
Find the median of the following grouped frequency distribution.
Length
Frequency
21 – 24
3
25 – 28
7
29 – 32
12
33 – 36
6
37 – 40
4
Cumulative
Frequency
Upper Limit
DH - Analysing Data
Cumulative
Frequency Curves
Median = Measure of central location
Q1 = ¼ (n + 1) = 8.25th → 26
Q2 = ½ (n +1) = 16.5th → 30
Q3 = ¾ (n +1) = 24.75th → 33
Interquartile Range = Q3 – Q1
= 33 – 26
=7
Q1 = 25th percentile
Q3 = 75th percentile
Cumulative frequency
Interquartile range = Measure of spread
= Q3 – Q 1
Q3
Q2
Q1
Upper Limit
DH - Analysing Data
Data Handling
Analysing Data
Learning Outcomes:
At the end of the topic I will be able to




Understand that in order to gain a mental picture of a
collection of data it is necessary to obtain a measure
of average and range
Be able to determine the mean, median and mode
for a set of raw scores and an ungrouped frequency
table
Be able to obtain the median and interquartile range
for grouped data from a cumulative frequency graph
each average and measure of spread
Can
Do
Revise
Further
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Data Handling
Presenting Data
Learning Outcomes
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Revise drawing of pie charts, line graphs and bar charts
Be able to present data using a stem and leaf diagram, determine
mean, Median and quartiles
Be able to draw a boxplot for a set of values and compare more than
one box and whisker plots with reference to their average, spread,
skewness
Be able to draw a histogram to represent groups with unequal widths
Know which diagram to use to represent data, the advantages and
Be aware of the shape of a normal distribution and understand the
concept of skewness
DH - Presenting Data
Box & Whisker Plots
A box & Whisker plot illustrates:
a) The range of data
b) The median of data
c) The quartiles and interquartile range of data
d) Any indication of skew within the data
Q1
Q2
Q3
Scale
Scatter Diagrams
DH - Presenting Data
y
× ×
×
× ×
x
Positive Correlation
x ▲ y▲
×
×
× ×
× ×
×
× ×
×
× ×
× ×
y
×
×
×
×
×
×
× ×
×
× ×
×
y
x
Negative Correlation
x ▲ y▼
x
No Correlation
x & y are independent
* The closer the points, the stronger the correlation
Histograms
DH - Presenting Data
32 packages were brought to the local post office. The masses of the packages
were recorded as follows
Mass (g)
0 < m ≤ 30
30 < m ≤ 40
40 < m ≤ 50
50 < m ≤ 90
No of packages
3
10
12
7
With unequal class widths we draw a histogram.
There are 2 important differences between a bar chart and a histogram
1. In a bar chart the height of the bar represents the frequency.
2. In a histogram the ‘x’ axis is a continuous scale.
Histograms
DH - Presenting Data
When the classes are of unequal width we calculate and plot frequency
density
Frequency Density = Frequency
Class Width
Group
Frequency
Class Width
0 < m ≤ 30
3
30
30 < m ≤ 40
10
10
40 < m ≤ 50
12
10
50 < m ≤ 90
7
40
Frequency
Density
Stem & Leaf Diagram
DH - Presenting Data
When data are grouped to draw a histogram or a cumulative frequency
distribution, individual results are lost. The advantage of grouping is that
patterns (distribution) can be seen. In a stem and leaf diagram individual
results are retained and the spread / distribution of the data can be seen.
Draw a stem and leaf diagram for the data:
10, 11, 12, 15, 23, 26, 29, 32, 33, 34, 35,36, 42, 43, 44, 56, 57
Stem
1
2
3
4
5
Leaf
DH - Presenting Data
Data Handling
Presenting Data






Can
Do
Revise
Further






Be able to draw a histogram to represent groups with
unequal widths


Know which diagram to use to represent data, the


Be aware of the shape of a normal distribution and
understand the concept of skewness


Revise drawing of pie charts, line graphs and bar charts
Be able to present data using a stem and leaf diagram,
determine mean, Median and quartiles
Be able to draw a boxplot for a set of values and
compare more than one box and whisker plots with
reference to their average, spread, skewness
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