Download measures of central tendency and dispersion

MEASURES OF CENTRAL
TENDENCY AND
DISPERSION AROUND
THE MEAN
Activity 1
1.
2.
3.
4.
e) 50,42857143
f) 7
Mean height = 182,2cm
Mean mass = 80kg
a) 1996: 13,27%
1997: 15,55%
1998: 19,07%
b) The percentage of women in SA infected
with HIV increased from 1996 to
1998.
a) 135 454,4444 km2
Measures of Dispersion around
the Mean
•
•
•
IQR is generally considered as a
measure of spread around the
median.
Variance and Standard Deviation
are measures of dispersion around
the mean.
A deviation from the mean
considers how far each element in
the data set differs from the mean.
•
•
•
Variance is the average of the
squares of the deviations of each
data item from the mean.
Large variance – data items are
widely spread.
Small variance – data items are
closely clustered around the mean.
Activity 2
1)
3)
a) 20 years
b) 20 years
Fred: variance = 4,67
Sipho: variance = 162,67
Standard Deviation
The Standard Deviation is the Square Root of
Variance.
Standard Deviation =
( x  x )
n
2
Activity 3
1.
11 years
2.
 x  x 
3.
= 196
n = 10
4,43 years (to two decimal places)
2
Using the Calculator to find
the Standard Deviation

To work out the Standard deviation for
Fred’s sister, press the following keys:
[MODE] [2:STAT]
[1: 1-VAR]
22 [=] 17 [=] 21 [=]
[AC]
[SHIFT] [1] (STAT)
[5: VAR]
[3: xn ]
[=] You should get 2,16
Now try for Sipho’s sisters.
This tells us that Sipho’s sisters ages are
more spread out than Fred’s.
Activity 4
1.
2.
a) (i) -6,6°c
(ii) 12,8°c
b) The temperatures are spread
out.
c)
a) 48,8 y
b) 10,3 y
c)
The Standard Deviation and
the Mean
[Hodge,S & SeeSons, Glasgow, page 78]
Activity 5
1.
2.
3.
Jabu: 12
Mmatsie: 12
Jabu: 5,2 (to one decimal place)
Mmatsie: 0,8 (to one decimal place)
Although these two students have the
same average, Mmatsie is more
consistent. Jabu does well in some
tests and badly in others.
Using Standard Deviations to
reach Conclusions
•
Provided that the sample size is reasonably
large and the data is not too skewed (that is,
it does not have some very large or very
small values), it is possible to make the
following approximate statements:
•
•
•
About 66% lie within one standard deviation of
the mean.
About 95% lie within two standard deviations of
the mean.
Almost all of the data will lie within three
standard deviations of the mean.
Activity 6
1.
2.
a) 22 calls
b) 5,3 calls
x  σ = 22±5,3
Interval is (16,7 ; 27,3)
The phone calls on Tues, Wed and Fri
fall within the interval.
3
 100%  60%
5