Median divides the area under the curve into two

Median divides the area under the curve into two equal halves
• Median is a resistant measure because it is more robust to extreme values or skewness than
the mean and therefore is a better measure of centre for a very skewed distribution
• Mean is not a resistant measure of centre, because it is seriously influenced by skewness
(pulled in the direction of a few extreme observations)
• Mode is the only measure of center that can also be used for qualitative data
• For skewed distributions, the best measure of center is the median
• For symmetric distributions, the best measure of center is the mean
Population Mean and Sample Mean
Population mean (μ) = summation of all items in the population
population size
1 2
i
N
y
y y y
N N

+ + +
= =

Where μ is the Greek letter “mew”
Subscript “i” indicates the i th observation
N = population size
Sample mean (
y
) = summation of all observations in a sample
sample size
1 2
i
n
y
y y y
y
n n
+ + +
= =

Where
y
is read “ y bar”
n = sample size

13
median
mode
mean
Symmetric distribution
(mode)
median
mean
Right-skewed Left-skewed
3.8 Measures of Variation (= spread)
Range = Max – Min = difference between the highest and lowest observations in a data set
• A biased measure of variation (because outliers can give a much wider range than is the real
spread of the main data set)
Sample Variance and Sample Standard Deviation
• Sample standard deviation is the best estimate of population standard deviation
Sample Variance and Sample Standard Deviation
Sample variance (s 2 ) = sum of squared deviations from the mean
Sample size – 1
( )
2
2
1
1
n
i
i
y y
s
n
=
−
=
−

Sample standard deviation (s) = positive square root of the sample variance
( )
2
2 sample variance
1
y y
s s
n
−
= = =
−


14
Example: Calculate the range, standard deviation and variance of the lengths (cm) of a sample of 5
gastropods of a certain species.
___________________________
1.4 0.8 0.9 1.3 0.7
___________________________
>>>>>>>>>>
Lengths (cm)
i
y
Deviation from
mean
( )
i
y y −
Squared
deviation
2
( )
i
y y −
1.4
0.8
0.9
1.3
0.7
Totals 5.1
>>>>>>>>>>
Degrees of Freedom (df)
• df = the number of independent observations
• For standard deviation, df = n – 1 because the sample mean is used as an estimate of the
population mean in calculating it (using defining formula). You first calculate the mean and
include that as part of the formula. When the sample mean is used and you know (n – 1)
observations, the n th observation is fixed and is therefore not independent.
• For standard deviation, df is the denominator of the formula
Population standard deviation and Population variance
Rounding Rules
1. Do not perform any rounding until all calculations are complete; otherwise substantial rounding
errors can occur.
2. When giving the final answer, keep at least one more decimal place than is given in the raw data.
Population standard deviation (

) (pronounced sigma)
2 ( )
i
y
N


−
=

Population variance = (population standard deviation)
2
=
2 

15
3.9 The Five-Number Summary and Boxplots
Percentiles – divide a data set into 100 equal parts
Deciles – divide a data set into 10 equal parts
Quartiles – divide a data set or distribution into 4 equal parts
25% 25% 25% 25%
Min Q
1
Q
2
Q
3
Max
First quartile (Q
1
) =