Measures of central tendency

The Measures of central tendency are values ​​with which a data set can be summarized or described. They are used to locate the center of a given data set.

It is called Measures of Central Tendency because generally the highest accumulation of data of a sample or population is in the intermediate values.

Commonly used Central Tendency Measures are:

Arithmetic average

Median

fashion

Central Tendency Measures in Ungrouped Data

Population: It is the total of elements that have a characteristic in common which is the object of an investigation.

Show: It is a representative subset of the population.

Ungrouped data: When the sample that has been taken from the population or process to be analyzed, that is, when we have at most 29 elements in the sample, then this data is analyzed in its entirety without the need to use techniques where the amount of work is reduced due to excess data.

Arithmetic average

It is symbolized by x ̅ and is obtained by dividing the sum of all values, between total observations. Its formula is:

x̅ = Σx / n

Where:

x = Are the values ​​or data

n = total number of data

Example:

The monthly commissions that a seller has received in the last 6 months are $ 9,800.00, $ 10,500.00, $ 7,300.00, $ 8,200.00, $ 11,100.00; $9,250.00. Calculate the Arithmetic Mean of the salary received by the seller.

x̅ = Σx / n

x̅ = (9800 + 10500 + 7300 + 8200 + 11100 + 9250) / 6

x̅ = $ 9,358.33

The average commission received by the seller is $ 9,358.33.

fashion

It is symbolized with (Mo) and it is the measure that indicates which data has the Highest Frequency in a data set, or which is repeated the most.

Examples:

1.- In the data set {20, 12, 14, 23, 78, 56, 96}

There is no repeating value in this data set, therefore this set of values Has no fashion.

2.- Determine the mode in the following set of data that correspond to the ages of girls in a kindergarten: {5, 7, 3, 3, 7, 8, 3, 5, 9, 5, 3, 4, 3} The most repeated age is 3, so so much, Fashion is 3.

Mo = 3

Median

It is symbolized by (Md) and it is the mean value of the data ordered in increasing order, it is the central value of a set of ordered values in increasing or decreasing form, and corresponds to the value that leaves the same number of values ​​before and after it in a data set grouped.

Depending on the number of values ​​you have, two cases can occur:

If he number of values ​​is odd, the Median will correspond to core value of that data set.

If he number of values ​​is even, the Median will correspond to average of the two central values (The core values ​​are added and divided by 2).

Examples:

1.- If you have the following data: {5, 4, 8, 10, 9, 1, 2}

When ordering them in increasing order, that is, from smallest to largest, we have:

{ 1, 2, 4, 5, 8, 9, 10 }

Md = 5 because it is the central value of the ordered set

2.- The following set of data is ordered in descending order, from highest to lowest, and corresponds to a set of even values, therefore, Md will be the average of the central values.

{ 21, 19, 18, 15, 13, 11, 10, 9, 5, 3 }

Md = (13 + 11) / 2

Md = 24/2

Md = 12

Central Tendency Measures in Grouped Data

When the data are grouped in Frequency Distribution Tables, the following formulas are used:

Arithmetic average

x̅ = Σ (fa) (mc) / n

Where:

fa = Absolute frequency of each class

mc = class mark

n = total number of data

fashion

Mo = Li + Ac [d₁ / (d₁+ d₂) ]

Where:

Li = Lower limit of the modal class

Ac = Width or class size

d₁ = Difference of the modal absolute frequency and the absolute frequency before that of the modal class

d₂ = Difference of the modal absolute frequency and the absolute frequency after that of the modal class.

The modal class is defined as one in which the absolute frequency is higher. Sometimes the modal class and the median class can be the same.

Median

Md = Li + Ac [(0.5n - fac) / fa]

Where:

Li = Lower limit of the middle class

Ac = Width or class size

0.5n = ½ n = total number of data divided by two

fac = cumulative frequency prior to that of the median class

fa = absolute frequency of the middle class

To define the median class, divide the total number of data by two. Subsequently, the accumulated frequencies are searched for the one that most closely approximates the result, if there are two equally approximate values ​​(lower and later), the lower one will be chosen.

Examples of Central Tendency Measures

1.- Calculate the Arithmetic Mean of the Data Set {1, 3, 5, 7, 9, 11, 13}

x̅ = Σx / n

x̅ = (1 + 3 + 5 + 7 + 9 + 11 + 13) / 7

x̅ = 49/7

x̅ = 7

2.- Detect the Mode of the Data Set {1, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 7, 9, 9, 11, 13, 13}

You have to see how many times each term of the set is listed

1: 1 time, 3: 2 times, 4: 3 times, 5: 4 times, 6: 3 times, 7: 1 time, 9: 2 times, 11: 1 time, 13: 2 times

Mo = 5, with 4 occurrences

3.- Find the Median of the Data Set {1, 3, 5, 7, 9, 11, 13}

There are 7 facts. The fourth data will have 3 data on the left and 3 data on the right.

{ 1, 3, 5, 7, 9, 11, 13 }

Md = 7, is the middle data

4.- Calculate the Arithmetic Mean of the Data Set {2, 4, 6, 8, 10, 12, 14}

x̅ = Σx / n

x̅ = (2 + 4 + 6 + 8 + 10 + 12 + 14) / 7

x̅ = 56/7

x̅ = 8

5.- Detect the Mode of the Data Set {2, 2, 2, 4, 4, 4, 6, 6, 6, 6, 6, 8, 8, 8, 10, 12, 14, 14}

You have to see how many times each term of the set is listed

2: 3 times, 4: 3 times, 6: 5 times, 8: 3 times, 10: 1 time, 12: 1 time, 14: 2 times

Mo = 6, with 5 occurrences

6.- Find the Median of the Data Set {2, 4, 6, 8, 10, 12, 14}

There are 7 facts. The fourth data will have 3 data on the left and 3 data on the right.

{ 2, 4, 6, 8, 10, 12, 14 }

Md = 8, is the middle data

7.- Calculate the Arithmetic Mean of the Data Set {3, 10, 14, 15, 19, 22, 35}

x̅ = Σx / n

x̅ = (3 + 10 + 14 + 15 + 19 + 22 + 35) / 7

x̅ = 118/7

x̅ = 16.85

8.- Detect the Mode of the Data Set {1, 3, 3, 4, 4, 4, 5, 6, 6, 6, 6, 6, 7, 9, 9, 11, 13, 13}

You have to see how many times each term of the set is listed

1: 1 time, 3: 2 times, 4: 3 times, 5: 1 time, 6: 5 times, 7: 1 time, 11: 1 time, 13: 2 times

Mo = 6, with 5 occurrences

9.- Find the Median of the Data Set {1, 9, 17, 25, 33, 41, 49}

There are 7 facts. The fourth data will have 3 data on the left and 3 data on the right.

{ 1, 9, 17, 25, 33, 41, 49 }

Md = 25, is the middle data

10.- Calculate the Arithmetic Mean of the Data Set {1, 9, 17, 25, 33, 41, 49}

x̅ = Σx / n

x̅ = (1 + 9 + 17 + 25 + 33 + 41 + 49) / 7

x̅ = 175/7

x̅ = 25