Measures of Central Tendency

Presented to: Dr. Naila Alam

Presented by: Sumia Syed

Umme Habiba

Sobia Idrees

Sajda Aish

Presented by: Sumia Syed

Central Tendency:Overview: In statistics, the

concept of an average, or representative, score is called Central Tendency.

The goal in measuring central tendency is to describe a distribution of scores by determining a single value that identifies the center of the distribution.

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• Ideally, this central value is the score that is the best representative value for all of the individuals in the distribution.

Continue..• In everyday language, central

tendency attempts to identify the ‘average’ or ‘typical’ individual. This average value can then be used to provide a simple distribution of an entire population or a sample.

• In addition, to describing an entire distribution, measures of central tendency are also useful for making comparisons between groups of individuals or between set of figures.

Methods: Measuring Central Tendency:

• Statisticians have developed three different methods for measuring central tendency;

1.The Mean

2.The Median and

3.The Mode

• Values called measures of central tendency are used to summarize data into a single value or statistic.– The mean is the sum of all the data

values divided by the number of values.– The median is the middle number when

the data are arranged in order.– The mode is the value that occurs most

frequently in the data.

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Continue..• A measure of central tendency is

a single value that attempts to describe a set of data by identifying the central position within that set of data. As such, measures of central tendency are sometimes called measures of central location.

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• They are also classed as summary statistics.

• The mean (often called the average) is most likely the measure of central tendency that you are most familiar with, but there are others, such as the median and the mode.

Examples of Measures of Central Tendency:

• For the data 1,2,3,4,5,5,6,7,8 the measures of central tendency are;

• Mean =

• Median = 5

• Mode = 5

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• The mean, median and mode are all valid measures of central tendency, but under different conditions, some measures of central tendency become more appropriate to use than others.

• In the following sections, we will look at the mean, median and mode, and learn how to calculate them and under what conditions they are most appropriate to be used.

MeanPresented by: Sobia idrees

M y topic Objectives are:

By the end of this presentation, you should be able to:

To identify the types of central tendencies.

Compute the mean of a given set of data.

Applications of mean.

Mean

The mean is the sum of the value of each observation in a dataset divided by the number of observations. This is also known as the arithmetic average.

The mean, often called the average, of a numerical set of data, is simply the sum of the data values divided by the number of values. The mean is the balance point of a distribution. The calculations for the mean of a sample and the total population are done in the same way. However, the mean of a population is constant, while the mean of a sample varies from sample to sample

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The mean of a sample or a population is computed by adding all of the observations and dividing by the number of observations.

Population mean = μ = ΣX / N     OR  Sample mean = x = Σx / n

where ΣX is the sum of all the population observations, N is the number of population observations, Σx is the sum of all the sample observations, and n is the number of sample observations.

Continued

When statisticians talk about the mean of a population, they use the Greek letter μ to refer to the mean score. When they talk about the mean of a sample, statisticians use the symbol x to refer to the mean score.

Group Data vs.Ungrouped Data

Group Data

Grouped data is when there is a large number of possible outcomes, we will usually need to group the data. For example: The ages of 200 people entering a park on a Saturday afternoon. The ages have been grouped into the classes 0-9, 10-19, 20-29, etc. 

Arithmetic Mean (group-data) :

Formula:Arithmetic Mean  = ΣfX/Σf where               X = Individual value               f = Frequency

The symbol of Σ is pronounced as sigma and is used to represent the sum of number.

Grouped Data Arithmetic Mean Example:

X -Value          Frequency(f)                 ΣfX1                      2                                  1 * 2 = 22                      3                                  2 * 3 = 63                      2                                  3 * 2 = 6                        Σf = 7                           ΣfX= 14   Step 1:          Find Σf.                         Σf = 7   Step 2:          Now, find ΣfX.                         ΣfX = 14   Step 3: Now, Substitute in the above formula given             Arithmetic mean = ΣfX/Σf = 14/7 = 2

Ungrouped data

Ungrouped data is the opposite of grouped data with only one possible answer.

For example: The ages of 200 people entering a park on a Saturday afternoon. The ages are: 27, 8, 10, 49 etc. 

Arithmetic Mean (ungroup-data)

Formula: Mean = sum of elements / number of elements                = a1+a2+a3+.....+an/n Arithmetic Mean = ΣX/n where                X = Individual value                n = Total number of values

Example: To find the mean of 3,5,7.   Step 1 Find the sum of the numbers.                         3+5+7 = 15   Step 2:       Calculate the total number.                         there are 3 numbers.   Step 3:       Finding mean.                        15/3 = 5                        Ans = 5

Shape of distribution Symmetrical distributions:

When a distribution is symmetrical, the mode, median and mean are all in the middle of the distribution.

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Skewed distributions:

When a distribution is skewed the mode remains the most commonly occurring value, the median remains the middle value in the distribution, but the mean is generally ‘pulled’ in the direction of the tails. In a skewed distribution, the median is often a preferred measure of central tendency, as the mean is not usually in the middle of the distribution. 

Positive skewed distribution:

A distribution is said to be positively or right skewed when the tail on the right side of the distribution is longer than the left side. In a positively skewed distribution it is common for the mean to be ‘pulled’ toward the right tail of the distribution. Positively mean > median

Negative skewed distribution:

A distribution is said to be negatively or left skewed when the tail on the left side of the distribution is longer than the right side. In a negatively skewed distribution, it is common for the mean to be ‘pulled’ toward the left tail of the distribution. Negatively mean < median

Advantages of Mean:

• It is easy to understand & simple calculate.• It is based on all the values.• It is easy to understand the arithmetic average even if

some of the details of the data are lacking.

Disadvantages of Mean:

• It is affected by extreme values.• It cannot be calculated for open end classes.

• It cannot be located graphically

• It gives misleading conclusions.

Median Presented by: Umm-e-Habiba

The Median

• The second measure of central tendencies

• The goal of the median is to locate the midpoint of the distribution.

• There are no specific symbols or notions to identify the median

Conti.…

• The median is simply identified by the word median.

• In addition, the definition and the computations for the median are identical for a sample and for a population.

The Definition

• If the scores in a distribution are listed in order from smallest to largest

• The median is the midpoint of the list

• More specifically, the median is the point on the measurement scale below which 50% of the scores in the distribution are located.

Finding the Median for most distribution

• The scores are divided into equal-sized group.

• We are not locating the midpoint from highest to lowest X values

• To find the median, list the scores in order from smallest to largest

Conti.…• Begin with the smallest score

and count the score as you move up the list

• The median is the first point you reach that is greater than of 50% of the score in the distribution

• The median can be equal to a score in the list or it can be a point between two scores

Conti.…

• Notice that the median is not algebraically defined (there is no equation for computing the median)

• Means that there is a degree of subjectivity in determining the exact value

Example

• This example demonstrates the calculation of the median when n is an “odd” number.

3, 5, 8, 10, 11

Example

•This example demonstrates the calculation of the median when n is an “even” number

1, 1, 4, 5, 7, 8

Example: Median of a set Grouped Data in a Distribution of Respondents by age

Age Group Frequency of Median class(f)

Cumulative frequencies(cf)

0-20 15 15

20-40 32 47

40-60 54 101

60-80 30 131

80-100 19 150

Total 150

Merits1. Simplicity

– It is very simple measure of central tendency

2. Free from the effect of extreme values

3. Real value

• Representative value as compared to arithmetic mean average, the value of which may not exist in the series at all.

Cont.…

Graphic presentation– It can be estimated also through the graphic

presentation of data.

Possible even when data is incomplete

Demerits

Unrealistic– When the median is located somewhere

between the two middle values, it remains only an approximate measure, not a precise value.

Cont.…

Lack of representative character– limited representative character as it is not

based on all the items in the series.

Cont.…

Lack of algebraic treatment– Arithmetic mean is capable of

further algebraic treatment, but median is not. For example, multiplying the median with the number of items in the series will not give us the sum total of the values of the series.

Mode Presented by: Sajda Aish

Mode

Mode is the most frequent value or

score in the distribution.

It is defined as that value of the item in

a series.

It is denoted by the capital letter Z.

Highest point of the frequencies

distribution curve.

Croxton and Cowden : defined it as “the mode of a distribution is the value at the point armed with the item tend to most heavily concentrated. It may be regarded as the most typical of a series of value”

The exact value of mode can be obtained by the following formula.

Z=L1

+

Monthly rent (Rs) Number of Libraries (f)

500-1000 5

1000-1500 10

1500-2000 8

2000-2500 16

2500-3000 14

3000 & Above 12

Total 65

Example: Calculate Mode for the distribution of monthly rent Paid by Libraries in Karnataka

Z=2000+

Z =2000+

Z=2400

Z=2000+0.8 ×500=400

Merits of Mode :• Mode is readily comprehensible

and easily calculated

• It is the best representative of data

• It is not at all affected by extreme value.

• The value of mode can also be determined graphically.

• It is usually an actual value of an important part of the series.

Demerits of Mode • It is not based on all

observations.

• It is not capable of further mathematical manipulation.

• Mode is affected to a great extent by sampling fluctuations.

• Choice of grouping has great influence on the value of mode.

Conclusion:A measure of central tendency is a measure that tells us where the middle of a bunch of data lies.Mean is the most common measure of central tendency. It is simply the sum of the numbers divided by the number of numbers in a set of data. This is also known as average.

Continue…Median is the number present in the middle when the numbers in a set of data are arranged in ascending or descending order. If the number of numbers in a data set is even, then the median is the mean of the two middle numbers.Mode is the value that occurs most frequently in a set of data.