Understanding Measures of Central Tendency and Dispersion, Study notes of English

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MEASURES OF CENTRAL TENDENCY

 Provides a very convenient way of describing a set of

scores with a single number that describes the

performance of a group.

 “ center ‘’ of the data

Measures of Central

Tendency

Properties

Mean

Refers to the arithmetic

average

• used when the data are in interval or in ratio level of

measurement

• used when the frequency distribution is regular, symmetrical,

or normal

• measures stability
• easily affected by extreme scores
• may not be an actual score in the distribution
• very easy to compute
• the sum of each score's distance from the mean is zero
• used to compute other measures such as standard deviation,

coefficient of variation, skewness, and z-score

EXAMPLE:

Mean is 4.285 or 4.

Measures of Central Tendency Properties

Mode

Refers to the score/s that occurs most frequently

in the distribution

• used when the data are in the

nominal level of measurement

• used when quick answer is

needed

• used when the score distribution

is normal

Types of mode

1. Unimodal is a score distribution that consists of one mode Ex: 2 + 2 + 10 + 5 + 5+ 5 + 1

2. Bimodal is a score distribution that consists of two modes. Ex: 2 + 2 + 3 + 3 + 10 + 1
3. Trimodal is a score distribution that consists of three modes. It is also considered as multi-modala score distribution that consists of more than two modes. Ex: 2 + 2 + 3 + 3 + 4 + 4 + 10 + 1
   • can be used for quantitative, as well as qualitative data
   • may not be unique
   • not affected by extreme values
   • may not exist at times

EXAMPLE:

Mode: 5

MEASURES OF VARIATION OR DISPERSION

• Is a single value that is used to describe the spread of the scores in

distribution

• Variation also known as “variability or dispersion

Measures of Variation or

Dispersion

Properties

INTER-QUARTILE RANGE (IQR)

refers to the distance between the

third quartile and the first quartile.

Interpretation:

 The larger the value of IQR, the

more dispersed the scores are from

the median value;

 The smaller the value of IQR, the

more clustered the scores are from

the median value.

• used when the data are in ordinal level of

measurement

• used when the frequency distribution is

irregular or skewed

• reduces the influence of the extreme scores

• considers only the middle 50% of the scores in

the distribution

• not easy to calculate as compared to the range

• the point of dispersion of the score is the

median value

LOWER

HALF

UPPER

HALF

Measures of Variation or

Dispersion

Properties

QUARTILE DEVIATION (QD) refers

to the average deviation of the third

quartile and the first quartile from the

value of the median.

Interpretation:

• The larger the value of QD, the

more dispersed the scores are from

the median value;

• The smaller the value of QD, the

more clustered the scores are from

the median value.

• used when the data are in ordinal level of

measurement

• used when the frequency distribution is

irregular or skewed

• reduces the influence of the extreme scores

• considers only the middle 50% of the scores in

the distribution

• not easy to calculate as compared to the range

• the point of dispersion of the score is the

median value

Measures of Variation or

Dispersion

Properties

STANDARD DEVIATION (S) refers to the average distance that deviates from the mean value. Interpretation:  If the value of standard deviation is large, on the average, the scores in the distribution will be far from the mean. Therefore, the scores are spread out around the mean value.T he distribution is also known as heterogeneous.  If the value of standard deviation is small, on the average, the scores in the distribution will be close to the mean. Hence, the scores are less dispersed or the scores in the distribution are homogeneous.

• used when the data are in interval or in ratio level

of measurement

• Used when the frequency distribution is regular,

symmetrical or normal

• The most important measure of variation or

dispersion

• The most commonly used measure of variation,

particularly in research

• Shows variation of the individual scores about the

mean

STANDARD DEVIATION – also know as the “square root of the variance” Population Variance Population Standard Deviation 𝝈 𝟐 = (𝑿 − 𝝁) 𝟐 𝑵 𝝈 = (𝑿 − 𝝁) 𝟐 𝑵 Sample Variance Sample Standard Deviation 𝒔 𝟐 = (𝑿 − 𝒙) 𝟐 𝒏 − 𝟏 𝒔^ =^ (𝑿 − 𝒙) 𝟐 𝒏 − 𝟏

MEASURES OF SKEWNESS

Describes the degree of departure of the scores from the symmetry. The skewness of s

score distribution only tells about the performance of the students, but nit reasons about

their performance.

Positively Skewed Negatively Skewed Normal Distribution

Skewed to the right; this

means that the thin end

tail of the curve goes to

the right part of the

distribution.

Skewed to the left ; meaning the

thin end tail of the curve goes to

the left part pf the distribution

The score is normally

distributed.

Most of the score are low ;

hence most of the students

got scores below the mean

value

Most of the scores are high, hence

most of the students got scores

above the mean value

It is symmetrical to the mean ;

the end tails to the curve can be

extended indefinitely in both

side and asymptotic to the

horizontal line

Mean value is greater

than the median and the

mode values

Ex: Mean=50;

Median=47;Mode=

Mean value is less than the

median and the mode

Ex: Mean=43;

Median=47;Mode=

The value of the mean, median

and mode are equal

Classification of Skewness and summary of their properties