Measures of Central Tendency and Variation

Central Tendency Meaning 

Central tendency is a statistical method that reflects a dataset’s central or usual value. It finds a single number that represents a whole probability distribution. These are the categories that describe what is average or typical of a given distribution. The mean, the mode, and the median are the three main measures of central tendency. They help in better understanding of central tendency meaning and central tendency formula. 

Different Measures Or Types Of Tendency

1. Mean

Mean is the most commonly used measure for measuring central tendency.

Mean is the sum of the value of each observation in a dataset divided by the total no. of observations.

There are four types of mean.

• Arithmetic Mean: It is nothing but the average of a dataset. It is calculated by adding all the values and then dividing them by the no. of observations.

             Formula: x-=∑x/n

• Weighted Mean: Weighted mean is calculated when some specific values are more important. A weight wi is attached to each value of xiwhen  xi is important.

Formula: Weighted Mean=   wxw

• Geometric Mean: It is the mean calculated using the product of the values instead of the sum. Here, the value is taken on the log scale.

Formula: Geometric Mean= n(x1)(x2)…(x3)

Log (GM)=∑(log x)/n

• Harmonic Mean: Harmonic mean is the reciprocal of the arithmetic mean once calculated. 

Formula: Harmonic Mean=1/(∑(1/x)/n) =n/∑(1/x)

2. Median

The median is the middle value in distribution when the values are arranged in an order, say ascending or descending order. It is used to measure central tendency when the datasheet is not huge.

In a distribution with an odd number of observations, the middle value is the median. When the distribution is with an even number of observations, the median is calculated by calculating the mean of the two middle values.

3. Mode

Mode is the most frequently occurring value in a distribution. It is possible to have more than one mode in a distribution. Such distributions are called bimodal or multimodal. 

Variation Meaning

In general terms, Variation means how much something differs from another. Variation gives us data about how that bata lies from the center of the distribution. This is important because the amount of variability indicates how well you can generalize results from the sample. Less variability means that you can better predict the information based on the sample dataset.

Ways to measure variation

• Range

The range is a statistical measure of dispersion, or how much the data is stretched out. It is the simple difference between the largest and the smallest value in a set of values.

Formula: Range= maximum value- minimum value

• Interquartile Range(IQR)

The interquartile range gives you the spread of the middle of your distribution.

For any distribution that’s ordered from low to high, the interquartile range contains half of the values. While the first quartile (Q1) contains the first 25% of values, the fourth quartile (Q4) contains the last 25% of values.The interquartile range is the third quartile (Q3) minus the first quartile (Q1). This gives us the range of the middle half of a data set.

• Variance

In a distribution, variance is the average squared deviation from the population mean. Variance reflects the degree of spread in the data set. The more spread the data, the larger the variance is concerning the mean.

Formula: Variance formula for sample

S2=(x-x )2N-1

Variance Formula for population

σ2=∑(X-μ)2/N

• Standard Deviation

Standard deviation is the mean of the variability of the distribution. It indicates on average, how far every single value lies from the mean. It is the square root of the variance.

Formula: σ=√σ2=∑(X-μ)2/N

Conclusion

While statistically analyzing any dataset about population or sample central tendency and variation are terms we come across. Measures that indicate the approximate center of distribution are called measures of central tendency. Measures that describe the spread of the data are measures of variation. These measures include the mean or average, median ( the middlemost value of an arranged dataset), mode (most frequently occurring value), range (the stretch of value in a dataset), variance (distance of each value from mean), and standard deviation.