Centeral Tendency

1. Measures of Central
Tendency
MEAN
MEDIAN
MODE
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2. Cont…
• Descriptive Statistics: Organize and Summarize a set of
scores.
• Central tendency is a statistical measure that attempts to
determine the single value, usually located in the center of a
distribution, that is most typical or most representative of the
entire set of scores.
describing an entire distribution (Sample or Population)
comparisons between groups (individuals or sets of figures)
• Number Crunching: A distribution of many scores is taken and
is “crunched” down to a single value that describes all the
scores.
• Problem: No single measure produces a central,
representative value in every situation.

4. Cont…
• The Mean, The Median, and The Mode are computed
differently and have different characteristics.
• To decide which of the three measures is best for any
particular distribution, keep in mind that the general purpose
of central tendency is to find the single most representative
score.

5. THE MEAN
• Arithmetic average, is computed by adding all the scores in the
distribution and dividing by the number of scores.
• mean for a population = μ
• mean for a sample = M (articles) or 𝑋 (Statistic Books)
• The mean for a distribution is the sum of the scores divided by the
number of scores.
• μ = ∑X/N
• M or 𝑋 = ∑X/n
• Dividing the total equally think of the mean as the amount each individual
receives when the total (∑X) is divided equally among all of the individuals.
• Demonstration 3.1 & 3.2
• The mean as a balance point for scores (1, 2, 6, 6, 10) each score
represented as a box that is sitting on a seesaw. If the seesaw is positioned
so that it pivots at a point equal to the mean, then it will be balanced and
will rest level.

7. Cont…
• Mean must be between lowest and highest value. If you
calculate a value that is outside this range, then you have
made an error.
• Combined Mean or Weighted Mean
•

8. Cont…
• Computing the mean from a frequency distribution table

9. Characteristics of the Mean
• Every score in the distribution contributes to the value of the
mean.
Changing a score: produces a new mean
Introducing a new score or removing a score changes the
mean with exception of new score (or the removed score)
exactly equal to the mean.
Adding or subtracting a constant from each score the same
constant is added to the mean
Multiplying or dividing each score by a constant the mean
changes in the same way.

11. THE MEDIAN
• The goal is to locate the midpoint of the distribution.
• If the scores in a distribution are listed in order from smallest
to largest, median is the point on the measurement scale
below which 50% of the scores in the distribution are located
i.e. scores are divided into two equal-sized groups.
• Simply identified by the word median, no symbol or notation
is used.
• Definition and computations are identical for a sample and for
a population.

12. Cont…
list the scores in order from smallest to largest.
Begin with the smallest score and count the scores as you
move up the list.
The median is the first point you reach that is greater than
50% of the scores in the distribution.
When n is an even number, locate the median by finding the
average of the middle two scores.
The median can be equal to a score in the list or it can be a point
between two scores.

13. • Median = [(n+1)/2]th Observation
• Median = l + h/f (n/2 – c)
• l: lower class limit of median class
• h: class interval/ height of the Class
• f: frequency of median class
• n: total number of observations/scores
• c: cumulative frequency of class preceding the median class

14. Characteristics of the Median
• Sum of absolute deviations of all values from their median is
always minimum than deviations from any other value.
• Median is less sensitive than mean to the presence of few
extreme scores (outliers).

15. Extension of Median
• Quartiles: The values that divide the distribution into four
equal parts so that one fourth of data fall under first quartile
(Q1) one half fall under second quartile (Q2) and three fourth
of data fall under third quartile (Q3).
• Q1 = [(n+1)/4]th Observation
• Q2 = 2 [(n+1)/4]th Observation
• Q3 = 3(n+1/4)th Observation
• Q1 = l + h/f (n/4 – c)
• Q2 = l + h/f (n/2 – c)
• Q3 = l + h/f (3n/4 – c)

16. Cont…
• Deciles: The values that divide the distribution into ten equal
parts and these are denoted by D1, D2, D3….D9.
• D1 = [(n+1)/10]th Observation
• D2 = 2 [(n+1)/10]th Observation
• D9 = 9 [(n+1)/10]th Observation
• D1 = l + h/f (n/10 – c)
• D2 = l + h/f (2n/10 – c)
• D9 = l + h/f (9n/10 – c)

17. Cont…
• Percentiles: The values that divide the distribution into 100
equal parts and these are denoted by P1, P2, P3….P99. It is such
a value that certain percentage of numbers fall below it and
rest of the numbers fall above it.
• Percentile ranks used in standardized test such as SAT e.g. raw
score may be 630 and percentile rank is 84 it means whoever
scored 630 has scored better than 84% of those who
underwent the exam. So 84 is not percentage it is the position
in the data.

18. Cont…
• P1 = [(n+1)/100]th Observation
• P2 = 2 [(n+1)/100]th Observation
• P99 = 99 [(n+1)/100]th Observation
• P1 = l + h/f (n/100 – c)
• P2 = l + h/f (2n/100 – c)
• P99 = l + h/f (99n/100 – c)

19. Mode
• In a frequency distribution, the mode is the score or category
that has the greatest frequency.
• It can be used to determine the typical value for any scale of
measurement.
• Mode is the only measure of central tendency that can be
used with data from a nominal scale of measurement.
• Only measure of central tendency that corresponds to an
actual score in the data (exception: adjacent values). Mean
and the median often produce an answer that does not equal
any score in the distribution.

20. Cont…
• A frequency distribution graph, the greatest frequency
appears at the tallest part of the figure.
• Simply identify the score located directly beneath the highest
point in the distribution.
• A distribution has only one mean and only one median, it is
possible to have more than one mode.
• A distribution with two modes is said to be bimodal, and a
distribution with more than two modes is called multimodal.
• A distribution with several equally high points is said to have
no mode.
• Major mode and minor mode

21. Selecting a Measure of Central
Tendency
• Mean is usually the preferred measure of central tendency whenever
the data consist of numerical scores as it uses every score in the
distribution.
• Added advantage of being closely related to variance and standard
deviation.
• Median is used when there are extreme scores or skewed distributions
since median is not affected by extreme scores.
• On occasions where there are undetermined values in the data set such
as the amount of time required for an individual to complete a task.
Since mean can’t be computed but median can.
• A distribution is said to be open-ended when there is no upper limit (or
lower limit) for one of the categories again mean can’t be computed but
median can provide with a representative value.
• When scores are measured on an ordinal scale, the median is always
appropriate and is usually the preferred measure of central tendency.
Since mean is defined in terms of distances, and because ordinal scales
do not measure distance, it is not appropriate to compute a mean.

22. Cont…
• Mode can be used to measure and describe central tendency
for data that are measured on a nominal scale. Because
nominal scales do not measure quantity (distance or
direction), it is impossible to compute a mean or a median.
• For discrete variables calculated means are usually fractional
values that cannot actually exist. So specifying the most
typical case is a preferred approach.
• It is used as a supplementary measure along with the mean or
median. This specification indicates the shape of the
distribution as well as a measure of central tendency.

23. Central Tendency and Shape Of the
Distribution
• There are situations in which all three measures have exactly the
same value. Likewise, there are situations in which the three
measures are guaranteed to be different. These situations are
determined by the shape of the distribution.
• For a perfectly symmetrical distribution with one mode, all three
measures of central tendency—the mean, the median, and the
mode—have the same value.
As each score on the left side of the distribution is balanced by a
corresponding score (the mirror image) on the right side.
Median is exactly at the center because exactly half of the area in
the graph is on either side of the center.
If a symmetrical distribution has only one mode, then it is also in
the center of the distribution.

25. Skewed Distributions
Positively Skewed Distribution
• Has the peak (highest frequency) on the left-hand side so it is
the position of the mode.
• Vertical line drawn at the mode does not divide the
distribution into two equal parts (median) must be located to
the right of the mode.
• Since mean is influenced most by the extreme scores in the
tail therefore it tends to be displaced toward the tail.

26. Cont…
Negatively Skewed Distributions
• Mode is the peak on right-hand side.
• Mean is displaced toward the left by the extreme scores in the
tail.
• Median is usually located between the mean and the mode.