Chapter 3: Central Tendency

• Central tendency:

• A statistical measure

• A single score that defines the center of a distribution

• Central tendency:

  • Purpose-

To find the single score that is most typical or most representative of the entire group

The Mean for a distribution is the:

Sum of ALL the scores Divided by the # of scores

• The Mean for a distribution 

  • Pop mean formula:

μ = (ΣX) / N

• The Mean for a distribution 

  • Sample mean formula:

M = (ΣX) / n

The value of the mean will always be located somewhere between the _____ score and the ____ score

highest & lowest

• The Weighted Mean:

  • Combine 2 sets of scores and then find the _____ mean for the combined group

• overall

• The Weighted Mean:

  • Combine 2 sets of scores and then find the _____ mean for the combined group

overall

• The Weighted Mean: 3 steps-

1. Determine the combined sum of ____the scores

2. Determine the combined ____ of scores

3. Divide the sum of scores by the ____numberof scores

1. ALL

2. #

3. total

• The Weighted Mean:

  • Formula-

M = (ΣX₁ + ΣX₂) / (n₁ + n₂)

Compute the sample mean from the table

M = ΣX / n

M = 66 / 8

• M = 8.25

A sample of n = 12 scores has a mean of M = 8. What is the value of ΣX for this sample?

M = ΣX / n

8 = ΣX / 12

• ΣX = 96

Does changing the value of any score change the mean?

Yes

Does introducing a new score or removing a score change the mean?

Yes

Is there any way that a mean does NOT change?

The score added or removed is exactly equal to the mean

• Adding or subtracting a constant from each score changes the mean by the _____ constant

• Multiplying or dividing each score by a constant __________ the mean by that constant

• same

• multiplies or divides

The Mean Is ________ ________ to Changes in Scores

Highly Sensitive

A sample of n = 7 scores has M = 5. All of the scores are doubled. What is the new mean?

5 X 2 = 10

The median is the ____ of the scores in a distribution when they are listed in order from smallest to largest

midpoint

The median divides the scores into two groups of ____ size

equal

Locate the Median:

• 5 11 3 10 8

median = 8

Locate the Median:

• 1 4 4 5 7 9

median = (4 + 5)/2 = 4.5

The mean is the _____ _____ of a distribution

balance point

• The mean is the balance point of a distribution

  • Defined by _______, so not the _____

• distances, center

Why is the mean much higher than the median in income earnings?

bc there are people that are EXTREMELY high earners

• True or False:

  • It is possible for more than 50% of the scores in a distribution to have values above the mean

• True

• True or False:

  • It is possible for more than 50% of the scores in a distribution to have values above the median

• False

The mode is the score or category that has the _______ _______ of any score in a frequency distribution

greatest frequency

Is it possible to have more than one mode?

• Yes

  • bimodal (and multimodal) distributions

• Symmetrical distributions:

  • _____ and _____ have same value

• Mean & Median

• Symmetrical distributions:

  • Mean & Median have same value

  • With ONE _____ it has the same value as the mean and the median

• mode

• A Perfectly Symmetrical distribution w/ ONE mode:

• Mean = Median = Mode

• Can a Symmetrical distribution have more than one mode?

• Can it have no modes?

• Yes

• Yes

Label

• Symmetrical Distribution!

  • Normal Distribution

Label

• Symmetrical Distribution!

  • Bimodal Distribution

Label

• Symmetrical Distribution!

  • Rectangular Distribution

• Skewed Distributions:

  • When mean is influenced by extreme scores, where is it found?

far toward the long tail (positive or negative)

• Skewed Distributions:

  • Is median found toward the long tail in Skewed Distributions?

Yes! but not as far as the mean

• Skewed Distributions:

  • Where is mode found in Skewed Distributions?

Near the short tail (the high point)

• Label

  • Measures of central tendency for positively and negatively skewed distributions

A distribution of scores shows the mean = 31 and the median = 43. This distribution is probably ____

negatively skewed

• Measure of Central Tendency:

  • Mean- Choose if you know the value of _____ score

Every

• Measure of central tendency:

  • Mean- Bad if

    • _______ scores

    • _______ & __________ Distribution

    • ________ values

    • ______ & _______ scale

• Extreme scores

• Skewed & Open-ended distribution

• Undetermined values

• Ordinal & Nominal scale

• Measure of Central Tendency:

  • Median- Choose if

    • _______ scores

    • _______ & __________ Distribution

    • ________ values

    • ______ scale

• Extreme scores

• Skewed & Open-ended distribution

• Undetermined values

• Ordinal scale

• Measure of Central Tendency:

  • Median- Bad if _______ scale

• Nominal Scale

• Measure of Central Tendency:

  • Mode- Choose if 

    • ______ scales

    • ______ variables

    • ______ shape

• Nominal scales

• Discrete variables

• Describing shape

• Measure of Central Tendency:

  • Mode- Bad if _____ or _____ data

• Interval or ratio data

• Do the median and talk about the outlier

  • The median is more informative about a typical household than the mean

• True or False:

  • The mean uses all of the scores in the data, so it is the best measure of central tendency for skewed data.

False

• True or False:

  • If the mean and median have the same values, the distribution is symmetrical

True