Central Tendency

1
Chapter 3

• Central Tendency

2
Concepts

• Central tendency
• Mean
• Median
• Mode

3
Mean

• Sum of scores divided by the number of scores.
• Population mean is represented by mu.

4
Example 1

• -Each car of a racing teams four cars was
  driven 23,000, 17,000, 9,000, and 13,000 miles.
  Find the average miles covered by each car.
• -Since this fleet is the population, the mean is
  (23,000 17,000 9,000 13,000)/415,500

5
Mean (cont)

• The sample mean is represented by

6
Mean example 2

• -A sample of five executives received bonuses
  last year of 14, 15, 17, 16, and 15 in 1000.
  Find the average bonus.
• -Since these is a sample of size 5, the sample
  mean is (14,000 15,000 17,000 16,000
  15,000)/5 15,400

7
Mean as a balance point for distribution

• The mean balances the distance of each score from
  the mean (like a seesaw).
• This helps determine how a distribution is
  affected if a new score is added or if an
  existing score is removed.
• See Fig 3.4

8
Properties of a mean

• Every set of interval-level and ratio-level data
  has a mean
• All the values are included in computing the mean
• A set of data has a unique mean

9
Weighted Mean

• When there are two samples (n1 n2) with
  corresponding means (X-bar1, X-bar2), how do you
  calculate the overall mean?
• Ex if n112, n27, X-bar16, X-bar27
• Weighted mean combined sum / combined n

10
Median

• Median The midpoint of the values after they
  have been ordered from the smallest to the
  largest, or the largest to the smallest. There
  are as many values above the median as below it
  in the data set.

11
Properties of a median

• There is a unique median for each data set.
• It is not affected by extremely large or small
  values
• and therefore it is a valuable measure of central
  tendency when such values occur)
• It can be computed for ratio-level,
  interval-level, and ordinal-level data.
• It can be computed for an open-ended frequency
  distribution if the median does not lie in an
  open-ended class.

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Notes about the median

• Note There are as many values above the median
  as below it in the data set.
• Note For an odd number of values, the median
  will be the middle value in the ordered set.
• Note For an even number of values, the median
  will be the arithmetic average of the teo middle
  values.

13
Median example 1

• Compute the median for The road life for a
  sample of five tires in miles is 42,000, 51,000,
  40,000, 39,000, and 48,000
• The data in ascending order is 39,000 40,000
  42,000 48,000 and 51,000. Thus the median is
  42,000 miles.

14
Median Example 2

• Compute the median for The years of service of a
  sample of six store managers is 16, 12, 8, 15, 7,
  and 23.
• Arranging in order gives 7,8,12,15,16,23. Thus
  the median is (1215)/2 13.5 years

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Median if there are several scores with the same
value in the middle of the distribution

• Ex 1, 2, 2, 3, 4, 4, 4, 4, 4, 5
• How would you normally calculate the median?
• Could use same method as last example, but this
  does not yield equal number of scores on the left
  side and right side of the histogram.
• See Fig 3.7
• Need to do an interpolation

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Interpolation steps for calculating the median

• Step 1 Count the number of scores (boxes) below
  the tied value
• Step 2 Find the number of additional scores
  (boxes) needed to make exactly one half of the
  total distribution
• Step 3 form a fraction
• Number of boxes needed (step 2)
• Number of tied boxes

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Interpolation (cont.)

• Step 4 Add the fraction (step 3) to the lower
  real limit of the interval containing the tied
  scores.
• Median 3.5 (0.5(10) 4)/5
• 3.5 (5-4)/5
• 3.5 1/5
• 3.5 0.2
• 3.7
• Can also use a cumulative frequency table to do
  an interpolation.

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Mode

• The mode is the value of the observation that
  appears most frequently (note there may be more
  than one mode for a data set Ex bimodal d).
• The exam scores for ten students are 81, 93, 75,
  68, 87, 81, 75, 81, 87. what is the modal exam
  score?
• Since the score of 81 occurs the most, then the
  modal score is 81.
• The mode can be used when the data is nominal.
• See table 3.5

19
Selecting a measure of central tendency

• Which central tendency measure you can use
  depends on the level of measurement your scores
  represent

20
Central tendency and the shape of the distribution

• See Fig. 316
• See Fig. 3.17