Download Grouped frequency Table

Measures of Central
Tendency of Grouped Data,
Standard Deviation,
Histograms and Frequency
Polygons
Univariate Data
• Concerned with a single attribute or
variable.
• Two forms of numerical data
– DISCRETE data – collected by counting
exact amounts
– CONTINUOUS data – values form part of
a continuous scale – generally collected
by measurement
Grouped frequency Table
•
•
•
•
For large volume of data.
Group data in frequency table.
Groups should not overlap.
Use inequality signs for continuous
data.
ORGANISING DATA
• Can organise data using
– a stem and leaf diagram/display
– A frequency table
– A grouped frequency table
• Data could be discrete or
continuous
Activity 1
1.
a)
Group
0 -10 hrs
tally
frequency
|||| |||| ||
12
11 – 20 hrs |||| |||| ||||
21 – 30 hrs ||||
14
4
TOTAL
30
b) How many learners watched less than 11
hours of TV a week?
How many learners watched more than 20
hours of TV a week?
2. a) Continuous data
b) Interval
Tally
1,50  x  1,55 ||
Frequency
2
1,55  x  1,60
|||
3
1,60  x  1,65
|||| |
6
1,65  x  1,70
|||| |
6
1,70  x  1,75 |||| ||
7
1,75  x  1,80 ||||
5
1,80  x  1,85 |||
3
1,85  x  1,90 |||
3
c) 11 learners
Measures of Central Tendency
from a Frequency Table
The Mean
• We use the following formula to find the
mean of ungrouped data:
x
where x
Mean =
x
=
,
n
= sum of the data items,
and n = the number of items
• We use a modified formula when finding the
mean of grouped data:
f . x
Mean = x = n , where f = the frequency,
n is the value of the item, andx = the
number of items
The median
The median is the middle data item when the
data is listed in order. We sometimes use the
n 1
formula 2 to find out which item is the
middle item, and can also find the median
from the frequency table.
The mode
The mode is the data item with the highest
frequency.
Mark obtained
x
Frequency
f
fx
0
10
0
1
20
20
2
40
80
3
50
150
4
30
120
5
30
150
6
20
120
7
20
140
8
10
80
9
10
90
10
10
100
n = 250
 f .x
= 1 050
• Mean = 4,2
• Median = 4
• Mode = 3
Activity 2
No. of
children
x
0
No. of
families
f
12
Total no. of
1. Mean no of
children
children
fx
per family
0
1
15
15
2
5
10
3
2
6
4
1
4
n = 35
Σfx = 35
=1
2. 18th
position, so
Median = 1
3. Mode =1
Measures of Central Tendency
from a Grouped Frequency Table
Total
frequency =
40.
Median is
20½th item.
So, median is
in interval:
8  h  10
Height (h) in
metres
2h  4
4h 6
6h 8
8  h  10
10  h  12
12  h  14
Frequency
2
6
11
12
8
1
n=40
This means
trees were
from 2 to 4 m.
This is the
group with the
highest
frequency in the
table. So, Modal
class is
8  h  10
Activity 3 ques 2
Height (in
metres)
Midpoint
Frequency
X
f
f .X
2h  4
3
2
6
4h 6
5
6
30
6h 8
8  h  10
7
11
77
9
12
108
10  h  12
12  h  14
11
8
88
13
1
13
n=40
∑ f . X =322
b) Mean ≈ 8,05
c) Median is 20,5th item.
Median is in the interval 8  h  10
Median ≈ 8  10  18  9
2
2
d) Modal class is 8  h  10
The Standard Deviation of Data
in a Frequency Table
To set up a frequency table press the
following keys:
[SHIFT] [SETUP]
Scroll down using ▼ arrow
[3:STAT] [1:ON]
Activity 4
3 a) Mean = 418g
b) Standard deviation = 21g
4. (418 ± 21) = (418 – 21; 418 + 21)
= (397;439)
6 cans lie in this interval.
4
 100% = 31%
13
Activity 5
50 < s ≤ 60
Midpoint of the
interval
X
55
60 < s ≤ 70
65
27
70 < s ≤ 80
75
25
80 < s ≤ 90
85
54
90 < s ≤ 100
95
21
100 < s ≤ 110
105
15
110 < s ≤ 120
115
8
120 < s ≤ 130
125
5
Speed in
km/h
No. of cars
f
20
2. a) Mean ≈ 14 435 ÷ 175 = 82 km/h
b) Median lies in 88th position, which is
between 80–89km/h.
This is approx 85 km/h.
i.e half of the drivers are driving at
85 km/h or more
3.
4. Standard deviation = 17,5
5.
(82 ± 17,5) = (82 – 17,5; 82 + 17,5)
= (64,5; 99,5)
~ 81%