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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