Central Value
It is often preferable to use a few numbers in order to summarize a distribution. One crucial context of distribution is where its centre is positioned. In this aspect, measures of central tendency are discussed primarily. A second important aspect of a distribution is how expanded it is. Simply to say, how much the data in the distribution differ from each other. Thus, a measure of central tendency represents the central value (centre point value) or typical value of a dataset.
Uses of Central Measures
Central tendencies are useful any time you want to summarize the central location of a dataset using a single value.
Finding a Central Value
The three most common measures of central tendency are as follows:
The mean
Median
Mode
Each of these measures computes the location of the central point using a different technique. Furthermore, these measures help to find out where most values in a distribution fall and are also known as the central location of a distribution.
Mean
The mean is the arithmetic average, and it is possibly the measure of central tendency that most of us are familiar with. The computation of the mean or average executes all values in the data. If you would change any value, the mean will also change. However, the mean doesn’t always trace the centre of the data or the central value correctly. The mean can be displayed in distributions.
Median
Median can be used for different measures of variability. If you’re not taking into account the mean because your data is skewed, you will see that using the median for the central tendency and interquartile range (IQR) for the variability goes together wonderfully. The median divides that data in half and the interquartile range tells you where the middle half of the data fall. The wider the IQR, the greater the spread the data spread. You can also use percentiles in order to identify the spread for other proportions. For example, 95% of the data fall between the 2.5th and 97.5th percentiles.
Mode
The mode is actually the component of the data set that occurs most frequently. One way to handle data spread when not using the mean is the mode. Mode for continuous data would be important when it comes to distributions that have two (bimodal) or more (multi-modal) peaks. In these cases, where one has more than one centre of tendency, it would seem that the mode measure of central tendency becomes the more important piece of information than either the mean or median.
What is Central Value?
When we define a central value, we state it to be a single value that attempts to define a data set by determining the central position within that set of data. Seeing that, a measure of central tendency is sometimes referred to as measures of central location or summary statistics.
Fun Facts on Central Value Statistics
By looking only on the mean, you don’t know how far from the mean any given observation is possible to fall.
If you knew an individual was male or female, those subpopulations will likely have different means.
The measure of central tendency cannot give a complete picture of data for interpretation.
Solved Examples on Central Measures
Example:
Palak's exam scores for her last chemistry class were: 67, 53, 74, 78. Find out the mean of these values.
Solution:
List the values of data set in order and divide the sum total of all values by the number of values in the data. Doing so, we get:
53 + 67 + 74 + 78 / 4
= 272/4
= 68
Note: if we get an answer in decimal form, then we round to one more decimal place than the initial data had.
Example: Find the median of the fluctuation in prices of onion: 4, 6, 3, 10, 8, 2, 5, 7, 6, 4, 8.
Solution:
Start by arranging the data in ascending order: 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, and 10.
Seeing that there are 11 data values, an even number, thus our middle number is 6.
Now, for finding a central value of the two middle numbers is 6 (6th value of the data set).
Thus, the median fluctuation in onion price is 6.
Conclusion: Although Mean, Median and Mode are the most commonly used methods of finding the central value of a number, there are other ways too like harmonic mean, geometric mean.
FAQs on Central Measures
1. What are measures of central tendency in statistics?
Measures of central tendency are single values that represent a dataset by identifying its central position. They help summarize the data and indicate where most values fall, providing a typical value for the set. The most common measures are the mean, median, and mode.
2. What are the main types of central measures used in data analysis?
The three primary measures of central tendency that are commonly used in data analysis are:
- Mean: This is the arithmetic average of all values. You calculate it by adding all numbers in the dataset and then dividing by the count of those numbers.
- Median: This is the middle value in a dataset after all values have been arranged in order from lowest to highest. If there’s an even number of data points, the median is the average of the two middle values.
- Mode: This is the value that appears most frequently in a dataset. A dataset can have one mode, multiple modes, or no mode at all.
3. How are the mean, median, and mode calculated for a given set of data?
For calculation:
- To find the mean, sum all the data points and divide the sum by the total number of data points.
- To determine the median, first arrange your data in either ascending or descending order. If the total number of data points is odd, the median is the single middle value. If it's even, the median is the average of the two middle values.
- To identify the mode, simply count how often each value appears. The value that occurs with the highest frequency is the mode.
4. Why are these measures called 'central tendencies'?
They are called 'central tendencies' because they describe the inclination or 'tendency' of data points to group around a central value. In many datasets, values tend to be concentrated around a specific point, and the mean, median, and mode each aim to locate this central position or typical value within the distribution.
5. When is it best to use the mean, median, or mode for a dataset?
The choice of which measure to use depends on the dataset and its purpose:
- The mean is generally the best choice for data that is symmetrically distributed without extreme values, as it uses all data points in its calculation.
- The median is preferred when your data contains outliers (extreme values) or is skewed (not symmetrical), because it is less affected by these unusual values.
- The mode is most useful for categorical data or when you need to identify the most frequent item, category, or response in a set, rather than a numerical average.
6. Can you explain the difference between mean, median, and mode with an example?
Let's use an example of student scores on a small quiz: 7, 8, 8, 9, 15.
- Mean: (7 + 8 + 8 + 9 + 15) / 5 = 47 / 5 = 9.4. This is the average score.
- Median: First, arrange the scores in order: 7, 8, 8, 9, 15. The middle value is 8.
- Mode: The score that appears most often is 8 (it appears twice).
As you can see, the values can differ. The high score of 15 pulled the mean up, while the median and mode remained closer to what feels like the 'typical' score.
7. How do extreme values (outliers) affect different measures of central tendency?
Outliers are data points that are significantly higher or lower than most other values in a dataset. Their impact varies:
- The mean is highly sensitive to outliers. A single extreme value can dramatically shift the mean away from the center of the majority of data points.
- The median is much more robust to outliers. Since it only relies on the position of the middle value(s), an extreme value at either end of the ordered list will have little to no effect on the median unless it changes which values are in the middle.
- The mode is generally not affected by outliers, as an outlier typically does not occur frequently enough to become the most common value in the dataset.
8. Where are measures of central tendency applied in real-life situations?
These measures are widely used in many everyday situations:
- In education, to calculate the average test score (mean) of a class or a student's performance.
- In business, to determine the most popular product size (mode) to produce, or to find the typical salary (median) in a company to avoid distortion from very high earners.
- In science and research, to summarize experimental data, like the average growth (mean) of plants under different conditions.
- In sports, to analyze player performance, such as the average points per game (mean) of a basketball player.
