Measures of location
Ľubica and Ján Krauskovi, Dominik Heger
Masaryk University hegerdfSchemi. muni, cz
STDT 03 Central tendencies
skovi, Dominik Heger (K
4 □ ►   4 S
STDT 03 Central tendencies
1 / 10
Examples of Distributions
Maternal ages
20 25 3D 35 43
iiaternal age (years)
skovi, Dominik Heger (K
4 □ ► <S
1 -o^o
STDT 03 Central tende
Examples of Distributions
Maternal ages
Birthweights of their babies
20 25 3D 35 43
iiaternal age (years)
100 '20 140 160 18C
birth weight iounces:
4 □ ► <S
1 -o^o
skovi, Dominik Heger (K
STDT 03 Central tende
A right-skewed distribution
Income in USA
0    10      25 50 10O 150
iiK:re ith: .-:r.r
iskovi, Dominik Heger (K Measures of location STDT 03 Central tendencies 3/10
Percentile
Formal definition of percentile
The ptn percentile of a list of numbers is the smallest number that is at least as large as p % of the list.
P-tý percentyl z řady čísel je takové nejmenší číslo, které je nejméně tak velké jako p % čísel v řadě.
iskovi, Dominik Heger (K
4 □ ►   4 S
STDT 03 Central tendencies 4/10
Percentile
Mechanism for median finding:
• order the list
• identify the place that is p % of the away from the bottom of the list
• if that's a place on the list, take the number in that place, if not, take the next one up
iskovi, Dominik Heger (K
4 □ ►   4 S
STDT 03 Central tendencies 5/10
Average
Average is a smoother.
iskovi, Dominik Heger (K Measures of location STDT 03 Central tendencies 6/10
Average
Average is a smoother.
1 n
» = -E
2^
4=1
skovi, Dominik Heger (K Measures of location STDT 03 Central tende
Average
Average is a smoother.
1 n
» = -E
4=1
1 v—"V 1 v-^ 2%
^ _ ^ Ex* _ E^'^^ _ E
4=1 4=1 4 = 1
skovi, Dominik Heger (K Measures of location STDT 03 Central tende
Comparing averages
age (years)	20-30	30-40	40-50	50-60	60-75	75+
average height(")	69.3	69.5	69.4	69.2	68.3	67.2
Intervals include the left endpoint but not the right. [National Health and Nutrition Examination Survey, 1999-2002]
iskovi, Dominik Heger (K
4 □ ►   4 S
STDT 03 Central tendencies 7/10
Comparing averages
age (years)	20-30	30-40	40-50	50-60	60-75	75+
average height(")	69.3	69.5	69.4	69.2	68.3	67.2
Intervals include the left endpoint but not the right. [National Health and Nutrition Examination Survey, 1999-2002]
Is this table telling us that as men get older, on average they get a bit taller and then get shorter?
iskovi, Dominik Heger (K
4 □ ►   4 S
STDT 03 Central tendencies 7/10
Comparing averages
age (years)	20-30	30-40	40-50	50-60	60-75	75+
average height(")	69.3	69.5	69.4	69.2	68.3	67.2
Intervals include the left endpoint but not the right. [National Health and Nutrition Examination Survey, 1999-2002]
Is this table telling us that as men get older, on average they get a bit
taller and then get shorter?
Data - longitudinal x cross-sectional
iskovi, Dominik Heger (K
4 □ ►   4 S
STDT 03 Central tendencies 7/10
Comparing averages
age (years)	20-30	30-40	40-50	50-60	60-75	75+
average height(")	69.3	69.5	69.4	69.2	68.3	67.2
Intervals include the left endpoint but not the right. [National Health and Nutrition Examination Survey, 1999-2002]
Is this table telling us that as men get older, on average they get a bit
taller and then get shorter?
Data - longitudinal x cross-sectional
The men in various categories are not the same.
iskovi, Dominik Heger (K
4 □ ►   4 S
STDT 03 Central tendencies 7/10
Comparing averages
age (years)	20-30	30-40	40-50	50-60	60-75	75+
average height(")	69.3	69.5	69.4	69.2	68.3	67.2
Intervals include the left endpoint but not the right. [National Health and Nutrition Examination Survey, 1999-2002]
Is this table telling us that as men get older, on average they get a bit
taller and then get shorter?
Data - longitudinal x cross-sectional
The men in various categories are not the same.
The older men were shorter, on average.
iskovi, Dominik Heger (K
4 □ ►   4 S
STDT 03 Central tendencies 7/10
Comparing averages
age (years)	20-30	30-40	40-50	50-60	60-75	75+
average height(")	69.3	69.5	69.4	69.2	68.3	67.2
Intervals include the left endpoint but not the right. [National Health and Nutrition Examination Survey, 1999-2002]
Is this table telling us that as men get older, on average they get a bit
taller and then get shorter?
Data - longitudinal x cross-sectional
The men in various categories are not the same.
The older men were shorter, on average.
When comparing averages first think: O How are the groups related to each other? O Take a look on the numerical averages.
iskovi, Dominik Heger (K Measures of location STDT 03 Central tendencies 7/10
What does an average tels us?
If a student's test score is above average, is the student in the top half of the class?
iskovi, Dominik Heger (K
4 □ ►   4 S
STDT 03 Central tendencies
8/10
What does an average tels us?
If a student's test score is above average, is the student in the top half of the class?
Not necessarily.
iskovi, Dominik Heger (K
4 □ ►   4 S
STDT 03 Central tendencies
8/10
Markov's inequality
If a list has only non-negative entries, then the proportion of entries that are at least as large as k times the average is a 1/k.
If a list has only non-negative entries, then the proportion of entries that are greater than or equal to k times the average is n 1/k.
Jestliže řada obsahuje pouze kladné členy, pak poměr členů, které jsou nejméně tak velké jako /c-krát průměr, je n 1/k.
iskovi, Dominik Heger (K
4 □ ►   4 S
STDT 03 Central tendencies
9/10
Infromation sources
https://courses.edx.org/courses/BerkeleyX/Stat_2.lx/
iskovi, Dominik Heger (K Measures of location STDT 03 Central tendencies        10 / 10