Central Tendency A statistical measure that serves as a descriptive statistic Determines a single value –summarize or condense a large set of data –accurately.

Central Tendency

• A statistical measure that serves as a descriptive statistic• Determines a single value

– summarize or condense a large set of data – accurately describes the center of the distribution– represents the entire distribution of scores

• Used to compare two (or more) sets of data– comparing the average score for one set versus the average score

for another set.

Figure 3-1Frequency distribution for ratings of attractiveness of a female face shown in a photograph for two groups of male participants: those who had consumed no alcohol and those who had consumed moderate amounts of alcohol.

Figure 3-2 (p. 74)Three distributions demonstrating the difficulty of defining central tendency. In each case, try to locate the “center” of the distribution

The Mean, the Median, and the Mode

• It is essential that central tendency be determined by an objective and well‑defined procedure so that others will understand exactly how the "average" value was obtained and can duplicate the process.

• No single procedure always produces a good, representative value.

• Therefore, researchers have developed three commonly used techniques for measuring central tendency:– the mean– the median– the mode

The Mean

• Most commonly used measure of central tendency • Requires scores on an interval or ratio scale• Computation

– sum, or total, for the entire set of scores – then dividing this sum by the number of scores

– Population formula ΣX/ N

– for example scores of 3, 7, 4, 6

• ΣX = 20 N = 4 20/4 = 5

– – Sample formula M = ΣX/ n or in some books

Figure 3-3 (p. 76)The frequency distribution shown as a seesaw balanced at the mean.

• Conceptually, the mean can also be defined as:• the amount that each individual receives when the total is divided

equally among all individuals • n = 6 boys with 180 baseball cards divided equally• M = ΣX/ n M = 180/6 = 30

• the balance point of the distribution

Weighted Mean

• When combining two sets of scores with different sample size – First Sample n = 12 ΣX = 72 M = 6– Second Sample n = 8 ΣX = 56 M = 7

• Overall Mean– WRONG (6+7)/2 = 6.5 only if samples are the same size– CORRECT (72 + 56) / (12 + 8) = 128/20 = 6.4– Combined sum divided by combined n

– M = ΣX/ n M = (ΣX1 + ΣX2 ) / n1 + n2

Table 3.1 (p. 78)Calculating the mean from a frequency distribution table. Statistics quiz scores for a section of n = 8 students.

ΣX = 66n = 8

M = 66/8 = 8.25

Changing the Mean

• Calculation of the mean involves every score in the distribution, so:– modifying a set of scores

• by discarding scores• by adding new scores• will usually change the value of the mean

• To determine how the mean will be affected – determine how the number of scores is affected– determine how the sum of the scores is affected

Figure 3-4 (p. 80) Adding a ScoreA distribution of n = 5 scores that is balanced with a mean of µ = 7. What if a new score X = 10 is added to the distribution?

Original samplen = 5ΣX =35M = 35/5 = 7

New samplen = 6ΣX = 45M = 45/6 = 8

10▼

→

.

.7

Changing the Mean

• Changing the value of any score will change the value of the mean – If constant value is added to every score in a

distribution,• then the same constant value is added to the mean

– If every score is multiplied by a constant value,• then the mean is also multiplied by the same constant value

Table 3.2 Adding or Subtracting a ConstantNumber of sentences recalled for humorous and nonhumorous sentences.

+2

Table 3.3 Multiplying or dividing by a constant; Measurement of five pieces of wood in inches and converted to centimeters. So mean centimeters is 2.54 times mean inches.

The Median• The midpoint of scores listed in order from smallest to

largest• 50% of the scores are equal to or less than the median • Same as the 50th percentile• Computation

– requires scores measured on an ordinal, interval, or ratio scale– simple counting procedure

• With an odd number of scores

– list the values in order– the median is the middle score in the list.

• With an even number of scores

– list the values in order– the median is half-way between the middle two scores

Example 3.5The median divides the area in the graph exactly in half.Scores of 3, 5, 8, 10, 11 organized by valueAn odd number of scores so the middle score is 8

Example 3.6 (modified)The median divides the area in the graph exactly in half.Scores of 3, 3, 4, 5, 7, 8 organized by valueAn even number of scores so the middle is between 4 and 5median is (4 + 5)/2 = 9 / 2 = 4.5

The Median

• If the scores are measurements of a continuous variable• it is possible to find the median by first placing the scores

in a frequency distribution histogram• with each score represented by a box in the graph. • Then, draw a vertical line through the distribution• so that exactly half the boxes are on each side of the line• The median is defined by the location of the line.

Figure 3-5 (page 84)A distribution with several scores clustered at the median. The median for this distribution is positioned so that each of the four boxes above X = 4 is divided into two sections, with 1/4 of each box below the median (to the left) and 3/4 of each box above the median (to the right). As a result, there are exactly four boxes, 50% of the distribution, on each side of the median.

3.5+0.25 4.5 – 0.75

Using a frequency distribution table to calculated 50% which is the median from example 3.7 numbersA continuous variable has upper and lower limits50% falls between 37.5 and 87.5 a distance of 5050% is 37.5 below 87.5% so 37.5/50 = 0.75distance between 3.5 and 4.5 is 1 1(0.75) = 0.75 4.5 – 0.75 = 3.75See box 3.2 on page 86

x f cf c%

6 1 8 100%

5 0 7 87.5%

4 4 7 87.5%

3 1 3 37.5%

2 1 2 25.0%

1 1 1 12.5%

Figure 3-6 (Page 86)A population of N = 6 scores with a mean of = 4.Notice that the mean does not necessarily divide the scores into two equal groups.In this example, 5 out of the 6 scores have values less than the mean. For these six scores 2, 2, 2, 3, 3, 12The median is the middle point in the scores in this case 2.5

Median

The Mode

• The most frequently occurring category or score in the distribution

• Peak in a frequency distribution graph• For data measured on any scale of measurement:

– nominal– ordinal– interval– ratio

Table 3.4 (p. 87)Favorite restaurants named by a sample of n = 100 students. Caution: The mode is a score or category, not a frequency. For this example, the mode is Luigi’s, not f = 42.

Bimodal Distributions

• It is possible for a distribution to have more than one mode. Such a distribution is called bimodal.

• In addition, the term "mode" is often used to describe a peak in a distribution that is not really the highest point. Thus, a distribution may have a major mode at the highest peak and a minor mode at a secondary peak in a different location.

Figure 3.7 (page 89) Bimodal distributionA frequency distribution for tone identification scores. An example of a bimodal distribution.

Selecting a Measure of Central Tendency

• Mean is preferred– uses every score in the distribution– commonly used in inferential statistics

• Situations where you cannot or should not compute a mean at all– nominal data– ordinal data (usually inappropriate)

• Situations where the mean does not provide a good, representative value– Extreme scores (see fig 3.8)

Figure 3-8 (p. 90) Frequency distribution of errors committed before reaching learning criterion. Example of effects of an extreme score on the meanThis is an obvious example but what if the scores where only a little skewed.Statistics to the rescue, there are tests for skewness

MeanM = ΣX/ n M = 203/10 = 20.3

Median is 11.5

Mode is 11.0

Selecting a Measure of Central Tendency

• Situations where the mean does not provide a good, representative value• Missing values

• Random Missing scores from errors, equipment failure ……• Usually remove all scores for that person• If a large number of scores are missing stop the research

• Undetermined values (see table 3.5)

• Open-ended distributions• For example: a score category of 5 or more pizzas• Can not calculate the mean • Plan ahead, try to get quantitative values

Table 3.5 (p. 91) Amount of time to complete puzzle.Undetermined values in the data set because Person 6 did not complete the puzzle. After 60 minutes the researcher stopped the test. There is no value for the 6th person so the mean can not be calculated.The Median 12.5 which between 3rd and 4th scores.

Note 1: some researchers record the maximum time referring to it as “timed out” in this case 60 which will be an extreme score instead of missing valueGenerally a bad idea even for experienced researchers because the value is really unknown.

Note 2: when it is one or two scores out of a set of one hundred scores some researchers treat this as random missing values and remove the person. i.e. remove #6. However, person #6 really did work on the puzzle and this person is part of the sample.

60

The Median• One advantage of the median is that it is relatively

unaffected by extreme scores. • The median tends to stay in the "center" of the distribution

even when– When the distribution is very skewed from a few extreme scores– Undetermined values; see table 3.5– Open-ended distribution

• Use the median for Ordinal measurement scale• In these situations, the median serves as a good alternative

to the mean.• Used as a supplemental measure of central tendency that is

reported along with the mean.

The Mode

• The only measure of central tendency that can be used for data measured on a nominal scale.

• Discrete variables are whole number– such as number of children in a family– Calculating the mean can produce fractions

• Families have 2.33 children

– Mode is more sensible but lacks accuracy• family has 2 children

• Used as a supplemental measure of central tendency that is reported along with the mean or the median.– Helps to describe shaped

Central Tendency and the Shape of the Distribution

• Mean, the median, and the mode are systematically related to each other.

• In a unimodal symmetrical distribution, the mode, mean, and median will all have the same value. (see fig 3.11)

• In a skewed distribution (see fig 3.12)– mode will be located at the peak on one side– the mean usually will be displaced toward the tail on the other

side.

• The median is usually located between the mean and the mode.

Figure 3-11 (p. 96)Measures of central tendency for three symmetrical distributions: normal, bimodal, and rectangular.

Figure 3-12 (p. 96) Measures of central tendency for skewed distributions.