For Any Data Set, Which Measures Of Central Location Have Only One Value?

Recommended: First read Measures of Shape

What are the measures of central tendency?

A measure of central tendency (also referred to as measures of centre or central location) is a summary measure that attempts to depict a whole set of data with a single value that represents the middle or eye of its distribution.

At that place are three chief measures of central trend: the manner, the median and the mean. Each of these measures describes a unlike indication of the typical or primal value in the distribution.

What is the style?

The fashion is the most ordinarily occurring value in a distribution.

Consider this dataset showing the retirement historic period of 11 people, in whole years: 54, 54, 54, 55, 56, 57, 57, 58, 58, 60, 60 This table shows a simple frequency distribution of the retirement age data.

Age	Frequency
54	3
55	1
56	i
57	2
58	2
60	ii

The well-nigh commonly occurring value is 54, therefore the mode of this distribution is 54 years. Advantage of the mode: The mode has an reward over the median and the hateful as it can be found for both numerical and categorical (non-numerical) information. Limitations of the manner: The are some limitations to using the way. In some distributions, the mode may not reflect the middle of the distribution very well. When the distribution of retirement age is ordered from lowest to highest value, information technology is easy to see that the centre of the distribution is 57 years, but the way is lower, at 54 years. 54, 54, 54, 55, 56, 57 , 57, 58, 58, 60, sixty It is too possible for there to be more i way for the same distribution of data, (bi-modal, or multi-modal). The presence of more than one mode tin can limit the ability of the mode in describing the center or typical value of the distribution considering a unmarried value to describe the eye cannot exist identified. In some cases, particularly where the data are continuous , the distribution may have no mode at all (i.e. if all values are different). In cases such every bit these, it may exist amend to consider using the median or hateful, or group the data in to appropriate intervals, and observe the modal grade.

What is the median?

The median is the middle value in distribution when the values are bundled in ascending or descending guild.

The median divides the distribution in one-half (there are 50% of observations on either side of the median value). In a distribution with an odd number of observations, the median value is the eye value. Looking at the retirement age distribution (which has 11 observations), the median is the middle value, which is 57 years: 54, 54, 54, 55, 56, 57 , 57, 58, 58, lx, threescore When the distribution has an even number of observations, the median value is the mean of the two middle values. In the following distribution, the two middle values are 56 and 57, therefore the median equals 56.5 years: 52, 54, 54, 54, 55, 56 , 57 , 57, 58, 58, 60, sixty Advantage of the median: The median is less affected past outliers and skewed data than the mean, and is ordinarily the preferred measure of central trend when the distribution is not symmetrical. Limitation of the median: The median cannot exist identified for chiselled nominal data, equally it cannot be logically ordered.

What is the hateful?

The hateful is the sum of the value of each observation in a dataset divided past the number of observations. This is as well known as the arithmetic average.

Looking at the retirement age distribution again: 54, 54, 54, 55, 56, 57, 57, 58, 58, 60, sixty The hateful is calculated past adding together all the values (54+54+54+55+56+57+57+58+58+threescore+60 = 623) and dividing by the number of observations (11) which equals 56.6 years. Advantage of the mean: The mean can be used for both continuous and discrete numeric information. Limitations of the mean: The hateful cannot be calculated for categorical data, every bit the values cannot be summed. As the mean includes every value in the distribution the mean is influenced by outliers and skewed distributions . What else practise I need to know well-nigh the hateful? The population mean is indicated by the Greek symbol � (pronounced 'mu'). When the hateful is calculated on a distribution from a sample it is indicated by the symbol x̅ (pronounced X-bar).

How does the shape of a distribution influence the Measures of Primal Trend?

Symmetrical distributions: When a distribution is symmetrical, the manner, median and mean are all in the eye of the distribution. The following graph shows a larger retirement age dataset with a distribution which is symmetrical. The manner, median and mean all equal 58 years.

Graph: Retirement age

Skewed distributions: When a distribution is skewed the mode remains the most commonly occurring value, the median remains the center value in the distribution, but the hateful is by and large '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 ordinarily in the middle of the 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 hateful to be 'pulled' toward the correct tail of the distribution. Although there are exceptions to this dominion, mostly, most of the values, including the median value, tend to exist less than the hateful value. The following graph shows a larger retirement historic period data gear up with a distribution which is right skewed. The data has been grouped into classes, as the variable being measured (retirement age) is continuous. The fashion is 54 years, the modal class is 54-56 years, the median is 56 years and the mean is 57.ii years.

Graph: Retirement age Positive (right) skew

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 hateful to be 'pulled' toward the left tail of the distribution. Although there are exceptions to this rule, generally, most of the values, including the median value, tend to exist greater than the mean value. The following graph shows a larger retirement age dataset with a distribution which left skewed. The mode is 65 years, the modal course is 63-65 years, the median is 63 years and the mean is 61.8 years.

Graph: Retirement age Negative (left) skew

How do outliers influence the measures of central trend?

Outliers are extreme, or atypical data value(s) that are notably different from the rest of the data.

It is important to find outliers within a distribution, because they can alter the results of the data assay. The mean is more than sensitive to the existence of outliers than the median or mode. Consider the initial retirement historic period dataset once again, with 1 difference; the last observation of lx years has been replaced with a retirement age of 81 years. This value is much higher than the other values, and could be considered an outlier. However, it has non inverse the eye of the distribution, and therefore the median value is still 57 years. 54, 54, 54, 55, 56, 57 , 57, 58, 58, 60, 81 As the all values are included in the adding of the mean, the outlier will influence the hateful value. (54+54+54+55+56+57+57+58+58+60+81 = 644), divided by 11 = 58.5 years In this distribution the outlier value has increased the mean value. Despite the beingness of outliers in a distribution, the hateful can withal be an appropriate measure of cardinal tendency, especially if the residual of the data is normally distributed. If the outlier is confirmed every bit a valid extreme value, it should non be removed from the dataset. Several common regression techniques can help reduce the influence of outliers on the mean value. Return to Statistical Language Homepage

Further information:

External links:

easycalculation.com - Hateful, Median, Mode Computer
calculatorsoup.com - Descriptive Statistics estimator
calculatorsoup.com - Hateful Median Fashion calculator