CENTRAL TENDENCY

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CENTRAL TENDENCY

• Mean, Median, and Mode

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OVERVIEW

• The general purpose of descriptive statistical
  methods is to organize and summarize a set score
• Perhaps the most common method for summarizing
  and describing a distribution is to find a single
  value that defines the average score and can
  serve as a representative for the entire
  distribution
• In statistics, the concept of an average or
  representative score is called central tendency

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OVERVIEW

• Central tendency has purpose to provide a single
  summary figure that best describe the central
  location of an entire distribution of observation
• It also help simplify comparison of two or more
  groups tested under different conditions
• There are three most commonly used in education
  and the behavioral sciences mode, median, and
  arithmetic mean

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The MODE

• A common meaning of mode is fashionable, and it
  has much the same implication in statistics
• In ungrouped distribution, the mode is the score
  that occurs with the greatest frequency
• In grouped data, it is taken as the midpoint of
  the class interval that contains the greatest
  numbers of scores
• The symbol for the mode is Mo

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The MEDIAN

• The median of a distribution is the point along
  the scale of possible scores below which 50 of
  the scores fall and is there another name for P50
• Thus, the median is the value that divides the
  distribution into halves
• It symbols is Mdn

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The ARITHMETIC MEAN

• The arithmetic mean is the sum of all the scores
  in the distribution divided by the total number
  of scores
• Many people call this measure the average, but we
  will avoid this term because it is sometimes used
  indiscriminately for any measure of central
  tendency
• For brevity, the arithmetic mean is usually
  called the mean

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The ARITHMETIC MEAN

• Some symbolism is needed to express the mean
  mathematically. We will use the capital letter X
  as a collective term to specify a particular set
  of score (be sure to use capital letters
  lower-case letters are used in a different way)
• We identify an individual score in the
  distribution by a subscript, such as X1 (the
  first score), X8 (the eighth score), and so forth
• You remember that n stands for the number in a
  sample and N for the number in a population

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Properties of the Mode

• The mode is easy to obtain, but it is not very
  stable from sample to sample
• Further, when quantitative data are grouped, the
  mode maybe strongly affected by the width and
  location of class interval
• There may be more than one mode for a particular
  set of scores. In rectangular distribution the
  ultimate is reached every score share the honor!
  For these reason, the mean or the median is often
  preferred with numerical data
• However, the mode is the only measure that can be
  used for data that have the character of a
  nominal scale

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Properties of the Median
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Properties of the Mean

• Unlike the other measures of central tendency,
  the mean is responsive to the exact position of
  reach score in the distribution
• Inspect the basic formula SX/n. Increasing or
  decreasing the value of any score changes SX and
  thus also change the value of the mean
• The mean may be thought of as the balance point
  of the distribution, to use a mechanical analogy.
  There is an algebraic way of stating that the
  mean is the balance point

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Properties of the Mean

• The sums of negative deviation from the mean
  exactly equals the sum of the positive deviation
• The mean is more sensitive to the presence (or
  absence) of scores at the extremes of the
  distribution than are the median or (ordinarily
  the mode
• When a measure of central tendency should reflect
  the total of the scores, the mean is the best
  choice because it is the only measure based of
  this quantity

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The MEAN of Ungrouped Data

• The mean (M), commonly known as the arithmetic
  average, is compute by adding all the scores in
  the distribution and dividing by the number of
  scores or cases

SX
M
N
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The MEAN of Grouped Data

• When data come to us grouped, or
• when they are too lengthy for comfortable
  addition without the aid of a calculating
  machine, or
• when we are going to group them for other purpose
  anyway,
• we find it more convenient to apply another
  formula for the mean

S f.Xc
M
N
X Xc f f.Xc
20 - 24 15 - 19 10 - 14 5 - 9 0 - 4
22 17 12 7 2
1 4 7 5 3
22 68 84 35 6
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The MEDIAN of Ungrouped Data

• Method 1 When N is an odd number
• ? list the score in order (lowest to highest),
  and the median is the middle score in the list
• Method 2 When N is an even number
• ? list the score in order (lowest to highest),
  and then locate the median by finding the point
  halfway between the middle two scores

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The MEDIAN of Ungrouped Data

• Method 3 When there are several scores with the
  same value in the middle of the distribution
• ? 1, 2, 2, 3, 4, 4, 4, 4, 4, 5
• There are 10 scores (an even number), so you
  normally would use method 2 and average the
  middle pair to determine the median
• By this method, the median would be 4

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f
f
5 4 3 2 1
5 4 3 2 1
X
X
0
0
1 2 3 4 5
1 2 3 4 5
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The MEDIAN of Grouped Data

• There are 10 scores (an even number), so you
  normally would use method 2 and average the
  middle pair to determine the median. By this
  method the median would be 4
• In many ways, this is a perfectly legitimate
  value for the median. However when you look
  closely at the distribution of scores, you
  probably get the clear impression that X 4 is
  not in the middle
• The problem comes from the tendency to interpret
  the score of 4 as meaning exactly 4.00 instead of
  meaning an interval from 3.5 to 4.5

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How to count the median?
0.5N f BELOW LRL
Mdn XLRL
f TIED
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THE MODE

• The word MODE means the most common observation
  among a group of scores
• In a frequency distribution, the mode is the
  score or category that has the greatest frequency

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SELECTING A MEASURE OF CENTRAL TENDENCY

• How do you decide which measure of central
  tendency to use? The answer depends on several
  factors
• Note that the mean is usually the preferred
  measure of central tendency, because the mean
  uses every score score in the distribution, it
  typically produces a good representative value
• The goal of central tendency is to find the
  single value that best represent the entire
  distribution

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SELECTING A MEASURE OF CENTRAL TENDENCY

• Besides being a good representative, the mean has
  the added advantage of being closely related to
  variance and standard deviation, the most common
  measures of variability
• This relationship makes the mean a valuable
  measure for purposes of inferential statistics
• For these reasons, and others, the mean generally
  is considered to be the best of the three measure
  of central tendency

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SELECTING A MEASURE OF CENTRAL TENDENCY

• But there are specific situations in which it is
  impossible to compute a mean or in which the mean
  is not particularly representative
• It is in these condition that the mode an the
  median are used

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WHEN TO USE THE MEDIAN

• Extreme scores or skewed distribution
• When a distribution has a (few) extreme
  score(s), score(s) that are very different in
  value from most of the others, then the mean may
  not be a good representative of the majority of
  the distribution.
• The problem comes from the fact that one or two
  extreme values can have a large influence and
  cause the mean displaced

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WHEN TO USE THE MEDIAN

• Undetermined values
• Occasionally, we will encounter a situation in
  which an individual has an unknown or
  undetermined score

Person Time (min.)
Notice that person 6 never complete the puzzle.
After one hour, this person still showed no sign
of solving the puzzle, so the experimenter stop
him or her
1 2 3 4 5 6
8 11 12 13 17 Never finished
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WHEN TO USE THE MEDIAN

• Undetermined values
• There are two important point to be noted
• The experimenter should not throw out this
  individuals score. The whole purpose to use a
  sample is to gain a picture of population, and
  this individual tells us about that part of the
  population cannot solve this puzzle
• This person should not be given a score of X 60
  minutes. Even though the experimenter stopped the
  individual after 1 hour, the person did not
  finish the puzzle. The score that is recorded is
  the amount of time needed to finish. For this
  individual, we do not know how long this is

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WHEN TO USE THE MEDIAN

• Open-ended distribution
• A distribution is said to be open-ended when
  there is no upper limit (or lower limit) for one
  of the categories

Number of children (X)
f
5 or more 4 3 2 1 0
3 2 2 3 6 4
Notice that is impossible to compute a mean for
these data because you cannot find SX
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WHEN TO USE THE MEDIAN

• Ordinal scale
• when score are measured on an ordinal scale, the
  median is always appropriate and is usually the
  preferred measure of central tendency

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WHEN TO USE THE MODE

• Nominal scales
• Because nominal scales do not measure quantity,
  it is impossible to compute a mean or a median
  for data from a nominal scale
• Discrete variables ? indivisible categories
• Describes shape
• the mode identifies the location of the peak
  (s). If you are told a set of exam score has a
  mean of 72 and a mode of 80, you should have a
  better picture of the distribution than would be
  available from mean alone

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CENTRAL TENDENCY AND THE SHAPE OF THE DISTRIBUTION

• Because the mean, the median, and the mode are
  all trying to measure the same thing (central
  tendency), it is reasonable to expect that these
  three values should be related
• There are situations in which all three measures
  will have exactly the same or different value
• The relationship among the mean, median, and mode
  are determined by the shape of the distribution

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SYMMETRICAL DISTRIBUTION SHAPE

• For a symmetrical distribution, the right-hand
  side will be a mirror image of the left-hand side
• By definition, the mean and the median will be
  exactly at the center because exactly half of the
  area in the graph will be on either side of the
  center
• Thus, for any symmetrical distribution, the mean
  and the median will be the same

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SYMMETRICAL DISTRIBUTION SHAPE

• If a symmetrical distribution has only one mode,
  it will also be exactly in the center of the
  distribution. All three measures of central
  tendency will have same value
• A bimodal distribution will have the mean and the
  median together in the center with the modes on
  each side
• A rectangular distribution has no mode because
  all X values occur with the same frequency. Still
  the mean and the median will be in the center and
  equivalent in value

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SYMMETRICAL DISTRIBUTION SHAPE
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POSITIVELY SKEWED DISTRIBUTION
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NEGATIVELY SKEWED DISTRIBUTION
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