2.1.1.2 Central Tendency | Median & Mode | Frequency | Grouped & Ungrouped data

In this video, you will get the idea of Central Tendency, Median & Mode, Frequency and Grouped & Ungrouped data.
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Median: Value of the variable which divides the whole distribution into two equal parts

Median class: The class in which the (𝑛/2)^𝑡ℎ cum. frequency falls. If 𝑛/2 is not present, 𝑛/2+1 will be considered to determine median class.

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Property 1: Number of the observations below and above the median is same.
Property 2: Median is not affected by extremely large or extremely small values or by open end class intervals.
When median is useful:
Useful when the distribution is skewed (asymmetric)

Merits of Median
1. Not affected by extreme values
2. Can be calculated for open end classes
3. Can be calculated even if the other classes are of unequal width

Demerits of Median
Not rigidly defined (a distribution may have more than one mode)
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1. For Ungrouped Data (Method 1)
If 𝑥_1,𝑥_2,…,𝑥_𝑛 are the 𝑛 observations then the arithmetic mean 𝑥 ̅=(〖𝑥_1+𝑥〗_2+…+𝑥_𝑛)/𝑛=(∑_(𝑖=1)^𝑛▒𝑥_𝑖 )/𝑛
If 𝑓_𝑖 is the frequency of 𝑢_𝑖their mean is 𝑥 ̅=(〖〖𝑓_1 𝑢〗_1+𝑓_2 𝑢〗_2+…+〖𝑓_𝑘 𝑢〗_𝑘)/(𝑓_1+𝑓_2+…+𝑓_𝑘 )=(∑_(𝑖=1)^𝑘▒〖𝑓_𝑖 𝑢〗_𝑖 )/(∑_(𝑖=1)^𝑘▒𝑓_𝑖 ), 𝑘 = Distinct observation count

1. For Ungrouped Data (Method 2)
Take deviations from any arbitrary point “A”. Then 𝑥 ̅=𝐴+(∑_(𝑖=1)^𝑛▒𝑑_𝑖 )/𝑛 𝑑_𝑖=𝑥_𝑖−𝐴
If 𝑓_𝑖 is the frequency of 𝑢_𝑖 then 𝑥 ̅=𝐴+(∑_(𝑖=1)^𝑘▒〖𝑓_𝑖 𝑑〗_𝑖 )/(∑_(𝑖=1)^𝑘▒𝑓_𝑖 )=(∑_(𝑖=1)^𝑘▒〖𝑓_𝑖 𝑢〗_𝑖 )/(∑_(𝑖=1)^𝑘▒𝑓_𝑖 )

2. For Grouped Data (Method 1)
If 𝑓_𝑖 is the frequency of 𝑚_𝑖 (the mid value of the 𝑖^𝑡ℎ class interval) their mean 𝑥 ̅=(〖〖𝑓_1 𝑚〗_1+𝑓_2 𝑚〗_2+…+〖𝑓_𝑘 𝑚〗_𝑘)/(𝑓_1+𝑓_2+…+𝑓_𝑘 )=(∑_(𝑖=1)^𝑘▒〖𝑓_𝑖 𝑚〗_𝑖 )/(∑_(𝑖=1)^𝑘▒𝑓_𝑖 ), 𝑘 = Distinct observation count.

2. For Grouped Data (Method 2)
If 𝐴 is an arbitrary point and 𝑓_𝑖 is the frequency of 𝑚_𝑖 (the mid value of the 𝑖^𝑡ℎ class interval) their mean
𝑥 ̅=𝐴+(〖〖𝑓_1 𝑑〗_1+𝑓_2 𝑑〗_2+…+〖𝑓_𝑘 𝑑〗_𝑘)/(𝑓_1+𝑓_2+…+𝑓_𝑘 )=𝐴+(∑_(𝑖=1)^𝑘▒〖𝑓_𝑖 𝑑〗_𝑖 )/(∑_(𝑖=1)^𝑘▒𝑓_𝑖 ), 𝑘 = Distinct observation count, 𝑑_𝑖=𝑚_𝑖−𝐴.

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