Level of Measurements in Statistics

Introduction to Level of Measurements in Statistics

Data can be classified according to the level of measurements in statistics, dictating the calculations that can be done to summarize and present the data (graphically), it also helps to determine, what statistical tests should be performed.

For example, suppose there are six colors of candies in a bag and you assign different numbers (codes) to them in such a way that brown candy has a value of 1, yellow 2, green 3, orange 4, blue 5, and red a value of 6. From this bag of candies, adding all the assigned color values and then dividing by the number of candies, yield an average value of 3.68. Does this mean that the average color is green or orange? Of course not. When computing statistic(s), it is important to recognize the data type, which may be qualitative (nominal and ordinal) and quantitative (interval and ratio).

The level of measurements in statistics has been developed in conjunction with the concepts of numbers and units of measurement. Statisticians classified measurements according to levels. There are four levels of measurement, namely, nominal, ordinal, interval, and ratio, described below.

Nominal Level of Measurement

At the nominal level of measurement, the observation of a qualitative variable can only be classified and counted. There is no particular order to the categories. Mode, frequency table (discrete frequency tables), pie chart, and bar graph are usually drawn for this level of measurement.

Ordinal Level of Measurement

In the ordinal level of measurement, data classification is presented by sets of labels or names that have relative values (ranking or ordering of values). For example, if you survey 1,000 people and ask them to rate a restaurant on a scale ranging from 0 to 5, where 5 shows a higher score (highest liking level) and zero shows the lowest (lowest liking level). Taking the average of these 1,000 people’s responses will have meaning. Usually, graphs and charts are drawn for ordinal data.

Level of Measurement

Interval Level of Measurement

Numbers also used to express the quantities, such as temperature, size of the dress, and plane ticket are all quantities. The interval level of measurement allows for the degree of difference between items but not the ratio between them. There is a meaningful difference between values, for example, 10 degrees Fahrenheit and 15 degrees is 5, and the difference between 50 and 55 degrees is also 5 degrees. It is also important that zero is just a point on the scale, it does not represent the absence of heat, just that it is a freezing point.

Ratio Level of Measurement

All of the quantitative data is recorded on the ratio level. It has all the characteristics of the interval level, but in addition, the zero points are meaningful and the ratio between two numbers is meaningful. Examples of ratio levels are wages, units of production, weight, changes in stock prices, the distance between home and office, height, etc.


Many of the inferential test statistics depend on the ratio and interval level of measurement. Many authors argue that interval and ratio measures should be named as scales.

Level of Measurements in Statistics

Importance of Level of Measurements in Statistics

Understanding the level of measurement in statistics, data is crucial for several reasons:

  • Choosing Appropriate Statistical Tests: Different statistical tests are designed for different levels of measurement. Using the wrong test on data with an inappropriate level of measurement can lead to misleading results and decisions.
  • Data Interpretation: The level of measurement determines how one can interpret the data and the conclusions can made. For example, average (mean) is calculated for interval and ratio data, but not for nominal or ordinal data.
  • Data analysis: The level of measurement influences the types of calculations and analyses one can perform on the data.

By correctly identifying the levels of measurement of the data, one can ensure that he/she is using appropriate statistical methods and drawing valid conclusions from the analysis.

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Median Definition, Formula, and Example: Quick Guide (2014)

Median Definition

Median (a measure of central tendency) is the middle-most value in the data set when all of the values (observations) in a data set are arranged either in ascending or descending order of their magnitude. The median is also considered a measure of central tendency that divides the data set into two halves, where the first half contains 50% observations below the median value and 50% above the median value. If there are an odd number of observations (data points) in a data set, the median value is the single-most middle value after sorting the data set.

After understanding the median definition, let us consider a few examples to calculate the median for a data set.

Median Example – 1

Question: For the following data set: 5, 9, 8, 4, 3, 1, 0, 8, 5, 3, 5, 6, 3, calculate the median.

Answer: To find the median of the given data set, first sort the data (either in ascending or descending order), that is
0, 1, 3, 3, 3, 4, 5, 5, 5, 6, 8, 8, 9. The middle-most value of the above data after sorting is 5, which is the median of the given data set.

When the number of observations in a data set is even then the median value is the average of two middle-most values in the sorted data.

Median Example – 2

Question: Consider the following data set, 5, 9, 8, 4, 3, 1, 0, 8, 5, 3, 5, 6, 3, 2. Compute the median.

Answer: To find the median first sort it and then locate the middle-most two values, that is,
0, 1, 2, 3, 3, 3, 4, 5, 5, 5, 6, 8, 8, 9. The middle-most two values are 4 and 5. So the median will be the average of these two values, i.e. 4.5 in this case.

The median is less affected by extreme values in the data set, so the median is the preferred measure of central tendency when the data set is skewed or not symmetrical.

Median Formula for Odd Number of Observations

For large data sets it is relatively very difficult to locate median values in sorted data. It will be helpful to use the median value using the formula. The formula for an odd number of observations is
$\begin{aligned}
Median &=\frac{n+1}{2}th\\
Median &=\frac{n+1}{2}\\
&=\frac{13+1}{2}\\
&=\frac{14}{2}=7th
\end{aligned}$

The 7th value in sorted data is the median of the given data.

Median Formula for Even Number of Observations

The median formula for an even number of observations is
$\begin{aligned}
Median&=\frac{1}{2}(\frac{n}{2}th + (\frac{n}{2}+1)th)\\
&=\frac{1}{2}(\frac{14}{2}th + (\frac{14}{2}+1)th)\\
&=\frac{1}{2}(7th + 8th )\\
&=\frac{1}{2}(4 + 5)= 4.5
\end{aligned}$

Median definition formula example

The computation of the median is a crucial step in exploratory data analysis (EDA). It helps identify potential outliers, assess skewness in the data distribution, and choose appropriate statistical methods for further analysis.

Applications of Median in Different Scenarios

1. Resisting Outliers: The median’s primary strength lies in its resistance to outliers. Unlike the mean (which can be swayed by extreme values), the median remains unaffected and stable by a few very high or very low data points (extreme observations).

2. Analyzing Skewed Distributions: When dealing with data that is not symmetrical (has skewed distributions), the median provides a more accurate representation of the “center” of the data compared to the mean/average. The median reflects the value that divides the data into halves, whereas the mean gets pulled towards the tail of the skewed distribution.

3. Ease of Interpretation: The median is a simple concept – the middle (centermost) value when the data is arranged in order (either ascending or descending).

Note that the median measure of central tendency, cannot be found for categorical data.

FAQs about Median

  1. What is the median?
  2. What is the advantage of the median over other measures of central tendencies?
  3. On what kind/type of data median can be computed?
  4. What is the benefit of using the median?
  5. Write the formula for the median when the number of observations is even and when the number of observations is odd.
  6. How median is interpreted?
  7. In how many groups median classify the data/sample/population?
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Mode Measure of Central Tendency (2014)

The mode is the most frequent observation in the data set i.e. the value (number) that appears the most in the data set. It is possible that there may be more than one mode or it may also be possible that there is no mode in a data set. Usually, it is calculated for categorical data (data belongs to nominal or ordinal scale) but is unnecessary.

It can also be used for ordinal and ratio scales, but there should be some repeated value in the data set or the data set can be classified. If any of the data points don’t have the same values (no repetition in data values), then the mode of that data set will not exit or may not be meaningful. A data set having more than one mode is called multimode or multimodal.

Example 1: Consider the following data set showing the weight of a child at the age of 10 years: 33, 30, 23, 23, 32, 21, 23, 30, 30, 22, 25, 33, 23, 23, 25. We can find the most repeated value by tabulating the given data in the form of a frequency distribution table, whose first column is the weight of the child and the second column is the number of times the weight appears in the data i.e. frequency of each weight in the first column.

Weight of 10 year childFrequency
221
235
252
303
321
332
Total15

From the above frequency distribution table, we can easily find the most repeated occurring observation (data point), which will be the mode of the data set and it is 23, meaning that the majority of the 10-year-old children weigh 23kg. Note that for finding the mode it is not necessary to make a frequency distribution table, but it helps in finding the mode quickly and the frequency table can also be used in further calculations such as percentage and cumulative percentage of each weight group.

Example 2: Consider we have information about a person about his/her gender. Consider the $M$ stands for male and $F$ stands for Female. The sequence of the person’s gender noted is as follows: F, F, M, F, F, M, M, M, M, F, M, F, M, F, M, M, M, F, F, M. The frequency distribution table of gender is

Weight of 10 year childFrequency
Male11
Female9
Total25

The most repeated gender is male, showing that the most frequent or majority of the people have male gender in this data set.

Mode can be found by simply sorting the data in ascending or descending order and then counting the frequent value without sorting the data especially when data contains a small number of observations, though it may be difficult to remember the number of times which observation occurs. Note that the mode is not affected by the extreme values (outliers or influential observations).

The mode is also a measure of central tendency, but it may not reflect the center of the data very well. For example, the mean of the data set in example 1, is 26.4kg while the mode is 23kg. Therefore, it should be used, if it is expected that data points will repeat or have some classification in them. For such kind of data, one should use it as a measure of central tendency instead of mean or median. For example,

  • In the production process, a product can be classified as a defective or non-defective product.
  • Student grades can classified as A, B, C, D, etc.
  • Gender of respondents
  • Blood Group

Example 3: Consider the following data. 3, 4, 7, 11, 15, 20, 23, 22, 26, 33, 25, 13. There is no mode of this data as each value occurs once. By grouping this data in some useful and meaningful form we can get the most repeated value of the data for example, the grouped frequency table is

GroupValuesFrequency
0 to 93, 4, 73
10 to 1911, 13, 153
20 to 2920, 22, 23, 25, 265
30 to 39331
Total12

We cannot find the most Frequent value from this table, but we can say that “20 to 29” is the group in which most of the observations occur. We can say that this group contains the mode which can be found by using the grouped formula.

Mode from Bar Graph

Bar Graph: Mode Measure of Central Tendency

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