## Recommended      ###### Measure of Central Tendency Statistics:- PDF / PPT

Measure of Central

Tendency

Statistics

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Measure of Central Tendency

• Central tendency is a statistical

measure that determines a single

value that accurately describes the

center of the distribution and

represents the entire distribution of

scores.

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Types of Averages

There are five common type, namely;

Arithmetic Mean (AM)

Median

Mode

Geometric Mean (GM)

Harmonic Mean (HM)

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Arithmetic Mean

• “The sum of all observations divide by the

total number of observation”.

𝑆𝑢𝑚 𝑜𝑓 𝑎𝑙𝑙 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛

Mean =

𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑎𝑙𝑙 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠

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Arithmetic Mean

AM for raw data

Find mean for the data

45, 32, 37, 46, 39, 36, 41, 48, 36

n=9

45+32+37+46+39+36+41+48+36

𝑥= ∙

𝑛

𝟑𝟔𝟎 Sample Population data

𝒙= = 40 data

𝟗 www.DuloMix.com n N

Arithmetic Mean

AM for Grouped data

Classes Frequency Mid Points (x) fx

68-87 10 77.5 775

88-107 13 97.5 1267.5

108-127 15 117.5 1762.5

128-147 9 137.5 1237.5

148-167 4 157.5 630

𝑓 = 51 𝑓𝑥 = 5402.5

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Median

• “A value which divides a data set that

have been ordered into two equal parts”.

OR

• “A median is a value at or below which

50% of ordered data lie”.

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Median

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Median

Median for raw data

For even size

Find Median for the data

0, 5, 3, 2, 6, 7

0, 2, 3, 5, 6, 7

n=6

𝑡ℎ 𝑡ℎ

1 6 6

𝑥 = The size of item + +1 item

2 2 2

o Symmetrical distribution

o Asymmetrical distribution

1. Symmetrical distribution

 The observations are equally distributed.

 The values of mean, median and mode are always equal.

i.e. Mean = Median = Mode

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Relations Among Averages

2. Asymmetrical distribution

 The observations are not equally distributed.

 Two possibilities are there:

Positively Skewed Negatively Skewed

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Quantiles

• When the number of observation is quite large, the

principle according to which a distribution or an

ordered data set is divided into two equal parts, may

be extended to any number of divisions.

• These are:

1. Quartiles

2. Deciles

3. Percentiles

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1. Quartiles

• “The three values which divide the

distribution into four equal parts”.

• These values are denoted by 𝑸𝟏, 𝑸𝟐 𝒂𝒏𝒅 𝑸𝟑.

• 𝑄1 𝑖𝑠 𝑐𝑎𝑙𝑙𝑒𝑑 𝑓𝑖𝑟𝑠𝑡 𝑜𝑟 𝑙𝑜𝑤𝑒𝑟 𝑞𝑢𝑎𝑟𝑡𝑖𝑙𝑒.

• 𝑄2 𝑖𝑠 𝑐𝑎𝑙𝑙𝑒𝑑 𝑠𝑒𝑐𝑜𝑛𝑑 𝑜𝑟 𝑚𝑖𝑑𝑑𝑙𝑒 𝑞𝑢𝑎𝑟𝑡𝑖𝑙𝑒.

• 𝑄3 𝑖𝑠 𝑐𝑎𝑙𝑙𝑒𝑑 𝑡ℎ𝑖𝑟𝑑 𝑜𝑟 𝑢𝑝𝑝𝑒𝑟 𝑞𝑢𝑎𝑟𝑡𝑖𝑙𝑒.

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Quartiles

𝑛

𝑸𝟏 = size of + 1 𝑡ℎ 𝒊𝒕𝒆𝒎

4

2𝑛

𝑸𝟐 = size of + 1 𝑡ℎ 𝒊𝒕𝒆𝒎

4

3𝑛

𝑸𝟑 = size of + 1 𝑡ℎ 𝒊𝒕𝒆𝒎

4

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2. Deciles

• “The nine values which divide

the distribution into ten equal

parts”.

• These values are denoted by 𝑫𝟏, 𝑫𝟐, … , 𝑫𝟗.

• Each Decile contains 10% of the total number

of observations.

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Deciles

𝑛 6𝑛

𝑫𝟏 = size of + 1 𝑡ℎ 𝒊𝒕𝒆𝒎 𝑫𝟔 = size of + 1 𝑡ℎ 𝒊𝒕𝒆𝒎

10 10

2𝑛

𝑫𝟐 = size of + 1 𝑡ℎ 𝒊𝒕𝒆𝒎 7𝑛

10 𝑫𝟕 = size of + 1 𝑡ℎ 𝒊𝒕𝒆𝒎

3𝑛 10

𝑫𝟑 = size of + 1 𝑡ℎ 𝒊𝒕𝒆𝒎

10 8𝑛

4𝑛 𝑫𝟖 = size of + 1 𝑡ℎ 𝒊𝒕𝒆𝒎

𝑫𝟒 = size of + 1 𝑡ℎ 𝒊𝒕𝒆𝒎 10

10

5𝑛 9𝑛

𝑫𝟓 = size of + 1 𝑡ℎ 𝒊𝒕𝒆𝒎 𝑫𝟗 = size of + 1 𝑡ℎ 𝒊𝒕𝒆𝒎

10 10

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3. Percentiles

• “The ninety nine values which divide

the distribution into hundred equal

parts”.

• These values are denoted by 𝑷𝟏, 𝑷𝟐, … , 𝑷𝟗𝟗.

• Each Decile contains 1% of the total number

of observations.

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