Statistics

1. MEASURES OF CENTRAL TENDENCY
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College of Nursing
3rd stage
Biostatistics
Dr. Nazar A. Mahmood

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Measures of Central Tendency:
• A measure of central tendency (also referred to as measures
of centre or central location) is a summary measure that
attempts to describe a whole set of data with a single value
that represents the middle or centre of its distribution.
• There are three main measures of central tendency: the
mode, the median and the mean.
• Each of these measures describes a different indication of
the typical or central value in the distribution.

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 Mode
 Median
 Mean
The choice of which measure to use depends on:
1. the shape of the distribution (whether normal or
skewed),
2. the variable‟s “level of measurement” (data are nominal,
ordinal or interval).

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Mode:
• The mode is the category or value that occurs most often (has the
heights frequency) in a set of data.
• It is the only measure of central tendency appropriate for nominal
data.
• Modal class is the category of nominal data with the greatest
frequency. For example, if a sample was composed of 50 nurses, 40
occupational therapists, 35 dental technicians, and 30 physical
therapists, the modal class would be nurses.
• The mode may also be reported for ordinal, interval, or ratio data. In
the following distribution of scores, the mode is 13 because it appears
twice and the other six numbers only appear once:
16 15 13 13 12 9 6 4

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Three types of mode:
• Unimodal: A frequency distribution that contains one
value that occurs more frequently than any other
• Bimodal: If two values have the same high frequency.
• Multimodal: If more than two values have the same high
frequency.

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Advantages of mode:
o It can be found for both numerical and categorical (non-numerical) data.
o It is easy to understand and simple to calculate.
o It is not affected by extreme large or small values.
o It can be located only by inspection in ungrouped data.
o Sometimes gives a clue about the etiology of the disease.
Disadvantages of mode:
o In some cases, particularly where the data are continuous, the
distribution may have no mode at all (i.e. if all values are different).
o In cases such as these, it may be better to consider using the median or
mean, or group the data in to appropriate intervals, and find the modal
class.

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B. Median (50th percentile):
• The median is the middle score or value in a group of data
• The median is appropriate for ordinal, interval, and ratio data.
• When ordinal data are analysed, the average category can be
identified. For example, if anxiety levels were classified as mild,
moderate, severe, and panic, the average category might be
moderate.
• With interval and ratio data, the median divides the frequency
distribution of the data in half.
• line up all the measured values in order from least to most or
from high to least, the value in the middle of the list is the
median.

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C. Mean:
• It is the most commonly known measure of central
tendency
• Arithmetic average used with interval or ratio.
• The population mean is indicated by the Greek symbol (µ)
(pronounced „mu‟). When the mean is calculated on a
distribution from a sample it is indicated by the symbol x̅
(pronounced X-bar).
• It is the sum of the values divided by the total number of
observations.

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Calculation of mean from grouped data

13. How does the shape of a distribution influence the
Measures of Central Tendency?
• Symmetrical distributions (Normal distribution):
When a distribution is symmetrical, the mode, median
and mean are all in the middle of the distribution.

14. Skewed distributions (asymmetrical):
When a distribution is skewed the mode remains the most commonly
occurring value, the median remains the middle value in the distribution, but
the mean is generally „pulled‟ in the direction of the tails. In a skewed
distribution, the median is often a preferred measure of central tendency,
as the mean is not usually in the middle of the distribution.

15. Relationship between the Three Measures of Mean, Median and Mode
1. For symmetric curve: Mean = Median = Mode
2. For positively skewed curve: Mean > Median > Mode
3. For negatively skewed curve: Mean < Median < Mode