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Obesity
Obesity is a medical condition in which excess body fat has accumulated to the extent that it may
have an adverse effect on health, leading to reduced life expectancy and/or increased health
problems. People are considered obese when their body mass index (BMI), a measurement
obtained by dividing a person's weight in kilograms by the square of the person's height
in metres, exceeds 30 kg/m2.
Classification

2. Obesity is a medical condition in which excess body fat has accumulated to the extent that it may
have an adverse effect on health. It is defined by body mass index (BMI) and further evaluated in
terms of fat distribution via the waist–hip ratio and total cardiovascular risk factors. BMI is
closely related to both percentage body fat and total body fat.
BMI Classification
< 18.5 underweight
18.5–24.9 normal weight
25.0–29.9 overweight
30.0–34.9 class I obesity
35.0–39.9 class II obesity
≥ 40.0 class III obesity
In children, a healthy weight varies with age and sex. Obesity in children and
adolescents is defined not as an absolute number but in relation to a historical
normal group, such that obesity is a BMI greater than the 95th percentile. The
reference data on which these percentiles were based date from 1963 to 1994,
and thus have not been affected by the recent increases in weight.
Measures of Central Tendency
Introduction
A measure of central tendency is a single value
that attempts to describe a set of data by identifying the central position within that set
of data. As such, measures of central tendency are sometimes called measures of
central location. They are also classed as summary statistics. The mean (often called
the average) is most likely the measure of central tendency that you are most familiar
with, but there are others, such as the median and the mode.
BMI is defined as the subject's mass divided
by the square of their height, expressed
kilograms per square meter and calculated as:

3. The mean, median and mode are all valid measures of central tendency, but under
different conditions, some measures of central tendency become more appropriate to
use than others. In the following sections, we will look at the mean, mode and median,
and learn how to calculate them and under what conditions they are most appropriate
to be used.
Mean
The mean (or average) is the most popular and well known measure of central
tendency. It can be used with both discrete and continuous data, although its use is
most often with continuous data (see our Types of Variable guide for data types). The
mean is equal to the sum of all the values in the data set divided by the number of
values in the data set. So, if we have n values in a data set and they have values x1, x2,
..., xn, the sample mean, usually denoted by (pronounced x bar), is:

4. This formula is usually written in a slightly different manner using the Greek capitol
letter, , pronounced "sigma", which means "sum of...":
You may have noticed that the above formula refers to the sample mean. So, why
have we called it a sample mean? This is because, in statistics, samples and
populations have very different meanings and these differences are very important,
even if, in the case of the mean, they are calculated in the same way. To acknowledge
that we are calculating the population mean and not the sample mean, we use the
Greek lower case letter "mu", denoted as µ:
The mean is essentially a model of your data set. It is the value that is most common.
You will notice, however, that the mean is not often one of the actual values that you
have observed in your data set. However, one of its important properties is that it
minimises error in the prediction of any one value in your data set. That is, it is the
value that produces the lowest amount of error from all other values in the data set.
An important property of the mean is that it includes every value in your data set as
part of the calculation. In addition, the mean is the only measure of central tendency
where the sum of the deviations of each value from the mean is always zero.
When not to use the mean
The mean has one main disadvantage: it is particularly susceptible to the influence of
outliers. These are values that are unusual compared to the rest of the data set by
being especially small or large in numerical value. For example, consider the wages of
staff at a factory below:

5. Staff 1 2 3 4 5 6 7 8 9 10
Salary 15k 18k 16k 14k 15k 15k 12k 17k 90k 95k
The mean salary for these ten staff is $30.7k. However, inspecting the raw data
suggests that this mean value might not be the best way to accurately reflect the
typical salary of a worker, as most workers have salaries in the $12k to 18k range. The
mean is being skewed by the two large salaries. Therefore, in this situation, we would
like to have a better measure of central tendency. As we will find out later, taking the
median would be a better measure of central tendency in this situation.
Another time when we usually prefer the median over the mean (or mode) is when our
data is skewed (i.e., the frequency distribution for our data is skewed). If we consider
the normal distribution - as this is the most frequently assessed in statistics - when the
data is perfectly normal, the mean, median and mode are identical. Moreover, they all
represent the most typical value in the data set. However, as the data becomes skewed
the mean loses its ability to provide the best central location for the data because the
skewed data is dragging it away from the typical value. However, the median best
retains this position and is not as strongly influenced by the skewed values. This is
explained in more detail in the skewed distribution section later in this guide.
Median
The median is the middle score for a set of data that has been arranged in order of
magnitude. The median is less affected by outliers and skewed data. In order to
calculate the median, suppose we have the data below:
65 55 89 56 35 14 56 55 87 45 92

6. We first need to rearrange that data into order of magnitude (smallest first):
14 35 45 55 55 56 56 65 87 89 92
Our median mark is the middle mark - in this case, 56 (highlighted in bold). It is the
middle mark because there are 5 scores before it and 5 scores after it. This works fine
when you have an odd number of scores, but what happens when you have an even
number of scores? What if you had only 10 scores? Well, you simply have to take the
middle two scores and average the result. So, if we look at the example below:
65 55 89 56 35 14 56 55 87 45
We again rearrange that data into order of magnitude (smallest first):
14 35 45 55 55 56 56 65 87 89 92
Only now we have to take the 5th and 6th score in our data set and average them to
get a median of 55.5.
Mode
The mode is the most frequent score in our data set. On a histogram it represents the
highest bar in a bar chart or histogram. You can, therefore, sometimes consider the
mode as being the most popular option. An example of a mode is presented below:

7. Normally, the mode is used for categorical data where we wish to know which is the
most common category, as illustrated below:
We can see above that the most common form of transport, in this particular data set,
is the bus. However, one of the problems with the mode is that it is not unique, so it
leaves us with problems when we have two or more values that share the highest
frequency, such as below:

8. We are now stuck as to which mode best describes the central tendency of the data.
This is particularly problematic when we have continuous data because we are more
likely not to have any one value that is more frequent than the other. For example,
consider measuring 30 peoples' weight (to the nearest 0.1 kg). How likely is it that we
will find two or more people with exactly the same weight (e.g., 67.4 kg)? The
answer, is probably very unlikely - many people might be close, but with such a small
sample (30 people) and a large range of possible weights, you are unlikely to find two
people with exactly the same weight; that is, to the nearest 0.1 kg. This is why the
mode is very rarely used with continuous data.
Another problem with the mode is that it will not provide us with a very good measure
of central tendency when the most common mark is far away from the rest of the data
in the data set, as depicted in the diagram below:

9. In the above diagram the mode has a value of 2. We can clearly see, however, that the
mode is not representative of the data, which is mostly concentrated around the 20 to
30 value range. To use the mode to describe the central tendency of this data set
would be misleading.
Summary of when to use the mean, median and mode

10. Please use the following summary table to know what the best measure of central
tendency is with respect to the different types of variable.
Type of Variable Best measure of central tendency
Nominal Mode
Ordinal Median
Interval/Ratio (not skewed) Mean
Interval/Ratio (skewed) Median
What is the best measure of central tendency?
There can often be a "best" measure of central tendency with regards to the data you
are analysing, but there is no one "best" measure of central tendency. This is because
whether you use the median, mean or mode will depend on the type of data you have,
such as nominal or continuous data; whether your data has outliers and/or is skewed;
and what you are trying to show from your data.
In a strongly skeweddistribution, what is the best indicator of central
tendency?
It is usually inappropriate to use the mean in such situations where your data is
skewed. You would normally choose the median or mode, with the median usually
preferred.
Does all data have a median, mode and mean?
Yes and no. All continuous data has a median, mode and mean. However, strictly
speaking, ordinal data has a median and mode only, and nominal data has only a
mode. However, a consensus has not been reached among statisticians about whether
the mean can be used with ordinal data, and you can often see a mean reported for
Likert data in research.
When is the mean the best measure of central tendency?

11. The mean is usually the best measure of central tendency to use when your data
distribution is continuous and symmetrical, such as when your data is normally
distributed. However, it all depends on what you are trying to show from your data.
When is the mode the best measure of central tendency?
The mode is the least used of the measures of central tendency and can only be used
when dealing with nominal data. For this reason, the mode will be the best measure of
central tendency (as it is the only one appropriate to use) when dealing with nominal
data. The mean and/or median are usually preferred when dealing with all other types
of data, but this does not mean it is never used with these data types.
When is the median the best measure of central tendency?
The median is usually preferred to other measures of central tendency when your data
set is skewed (i.e., forms a skewed distribution) or you are dealing with ordinal data.
However, the mode can also be appropriate in these situations, but is not as commonly
used as the median.
What is the most appropriate measure of central tendency when the data
has outliers?
The median is usually preferred in these situations because the value of the mean can
be distorted by the outliers. However, it will depend on how influential the outliers
are. If they do not significantly distort the mean, using the mean as the measure of
central tendency will usually be preferred.
In a normally distributed data set,which is greatest: mode, median or
mean?
If the data set is perfectly normal, the mean, median and mean are equal to each other
(i.e., the same value).

12. Before you can begin to understand statistics, there are four terms you will need to
fully understand. The first term 'average' is something we have been familiar with
from a very early age when we start analyzing our marks on report cards. We add
together all of our test results and then divide it by the sum of the total number of
marks there are. We often call it the average. However, statistically it's the Mean!
The Mean
Example:
Four tests results: 15, 18, 22, 20
The sum is: 75
Divide 75 by 4: 18.75
The 'Mean' (Average) is 18.75
(Often rounded to 19)
The Median
The Median is the 'middle value' in your list. When the totals of the list are odd, the
median is the middle entry in the list after sorting the list into increasing order. When
the totals of the list are even, the median is equal to the sum of the two middle (after
sorting the list into increasing order) numbers divided by two. Thus, remember to line
up your values, the middle number is the median! Be sure to remember the odd and
even rule.
Examples:
Find the Median of: 9, 3, 44, 17, 15 (Odd amount of numbers)
Line up your numbers: 3, 9, 15, 17, 44 (smallest to largest)
The Median is: 15 (The number in the middle)
Find the Median of: 8, 3, 44, 17, 12, 6 (Even amount of numbers)
Line up your numbers: 3, 6, 8, 12, 17, 44
Add the 2 middles numbers and divide by 2: 8 12 = 20 ÷ 2 = 10
The Median is 10.

13. The Mode
The mode in a list of numbers refers to the list of numbers that occur most frequently.
A trick to remember this one is to remember that mode starts with the same first two
letters that most does. Most frequently - Mode. You'll never forget that one!
Examples:
Find the mode of:
9, 3, 3, 44, 17 , 17, 44, 15, 15, 15, 27, 40, 8,
Put the numbers is order for ease:
3, 3, 8, 9, 15, 15, 15, 17, 17, 27, 40, 44, 44,
The Mode is 15 (15 occurs the most at 3 times)
*It is important to note that there can be more than one mode and if no number occurs
more than once in the set, then there is no mode for that set of numbers.
Occasionally in Statistics you'll be asked for the 'range' in a set of numbers. The range
is simply the the smallest number subtracted from the largest number in your set.
Thus, if your set is 9, 3, 44, 15, 6 - The range would be 44-3=41. Your range is 41.
A natural progression once the 3 terms in statistics are understood is the concept of
probability. Probability is the chance of an event happening and is usually expressed
as a fraction. But that's another topic!

14. What is statistics?
Statistics is the practice or science of collecting and analyzing numerical data in large
quantities.
History
Statistical methods date back at least to the 5th century BC. The earliest known
writing on statistics appears in a 9th-century book entitled Manuscript on Deciphering
Cryptographic Messages, written by Al-Kindi. In this book, Al-Kindi provides a
detailed description of how to use statistics and frequency analysis to decipher
encrypted messages. This was the birth of both statistics and cryptanalysis, according
to the Saudi engineer Ibrahim Al-Kadi.
The Nuova Cronica, a 14th-century history of Florence by the Florentine banker and
official Giovanni Villani, includes much statistical information on population,
ordinances, commerce, education, and religious facilities, and has been described as
the first introduction of statistics as a positive element in history.
Some scholars pinpoint the origin of statistics to 1663, with the publication of Natural
and Political Observations upon the Bills of Mortality by John Graunt. Early
applications of statistical thinking revolved around the needs of states to base policy
on demographic and economic data, hence its stat-etymology. The scope of the
discipline of statistics broadened in the early 19th century to include the collection
and analysis of data in general. Today, statistics is widely employed in government,
business, and natural and social sciences.
Its mathematical foundations were laid in the 17th century with the development of
the probability theory by Blaise Pascal and Pierre de Fermat. Probability theory arose
from the study of games of chance. The method of least squares was first described
by Carl Friedrich Gauss around 1794. The use of modern computers has expedited

15. large-scale statistical computation, and has also made possible new methods that are
impractical to perform manually.
Important contributors to statistics
Thomas Bayes George E. P. Box Pafnuty Chebyshev
David R. Cox Harald Cramér Francis Ysidro
Edgeworth
Gertrude Cox Bradley Efron Bruno de Finetti
Ronald A. Fisher Francis Galton Carl Friedrich Gauss
William Sealey Gosset Andrey Kolmogorov Pierre-Simon Laplace
Erich L. Lehmann Aleksandr Lyapunov Abraham De Moivre
Jerzy Neyman Florence Nightingale Blaise Pascal
Karl Pearson Charles S. Peirce Adolphe Quetelet
C. R. Rao Walter A. Shewhart Charles Spearman
Charles Stein Thorvald N. Thiele John Tukey
Abraham Wald