# Descriptive statistics by hand

# Introduction

This article explains how to compute the main descriptive statistics by hand and how to interpret them. To learn how to compute these measures in R, read the article “Descriptive statistics in R”.

Descriptive statistics (in the broad sense of the term) is a branch of statistics aiming at summarizing, describing and presenting a series of values or a dataset. Long series of values without any preparation or without any summary measures are often not informative due to the difficulty of recognizing any pattern in the data. Below an example with the height (in cm) of a population of 100 adults:

*188.7*, *169.4*, *178.6*, *181.3*, *179*, *173.9*, *190.1*, *174.1*, *195.2*, *174.4*, *188*, *197.9*, *161.1*, *172.2*, *173.7*, *181.4*, *172.2*, *148.4*, *150.6*, *188.2*, *171.9*, *157.2*, *173.3*, *187.1*, *194*, *170.7*, *172.4*, *157.4*, *179.6*, *168.6*, *179.6*, *182*, *185.4*, *168.9*, *180*, *157.8*, *167.2*, *166.5*, *150.9*, *175.4*, *177.1*, *171.4*, *182.6*, *167.7*, *161.3*, *179.3*, *166.9*, *189.4*, *170.7*, *181.6*, *178.2*, *167.2*, *190.8*, *181.4*, *175.9*, *177.8*, *181.8*, *175.9*, *145.1*, *177.8*, *171.3*, *176.9*, *180.8*, *189*, *167.7*, *188*, *178.4*, *185.4*, *184.2*, *182.2*, *164.6*, *174.1*, *181.2*, *165.5*, *169.6*, *180.8*, *182.7*, *179.6*, *166.1*, *164*, *190.1*, *177.6*, *175.9*, *173.8*, *163.1*, *181.1*, *172.8*, *173.2*, *184.3*, *183.2*, *188.9*, *170.2*, *181.5*, *188.9*, *163.9*, *166.4*, *163.7*, *160.4*, *175.8* and *181.5*

Facing this series, it is hard (not to say impossible) for anyone to understand the data and have a clear view of the size of these adults in a reasonable amount of time. Descriptive statistics allow to summarize, and thus have a better overview of the data. Of course, by summarizing data through one or several measures, some information will inevitably be lost. However, in many cases it is generally better to lose some information but in return gain an overview.

Descriptive statistics is often the first step and an important part in any statistical analysis. It allows to check the quality of the data by detecting potential outliers (i.e., data points that appear to be separated from the rest of the data), collection or encoding errors. It also helps to “understand” the data and if well presented, descriptive statistics is already a good starting point for further analyses.

# Location versus dispersion measures

Several different measures (called statistics if we are analyzing a sample) are used to summarize the data. Some of them give an understanding about the location of the data, others give and understanding about the dispersion of the data. In practice, both types of measures are often used together in order to summarize the data in the most concise but complete way. We illustrate this point with the graph below, representing the height (in cm) of 100 persons divided into two groups (50 persons in each group):

The black line corresponds to the mean. The mean height (in cm) is similar in both groups. However, it is clear that the dispersion of heights are very different in the two groups. For this reason, location or dispersion measures are often not enough if presented individually and it is a good practice to present several statistics from both types of measures.

In the following sections, we detail the most common location and dispersion measures and illustrate them with examples.

# Location

Location measures allow to see “where” the data are located, around which values. In other words, location measures give an understanding on what is the central tendency, the “position” of the data as a whole. It includes the following statistics (others exist but we focus on the most common ones):

- minimum
- maximum
- mean
- median
- first quartile
- third quartile
- mode

We detail and compute by hand each of them in the following sections.

## Minimum and maximum

Minimum (\(min\)) and maximum (\(max\)) are simply the lowest and largest values, respectively. Given the height (in cm) of a sample of 6 adults:

*188.7*, *169.4*, *178.6*, *181.3*, *179* and *173.9*

The minimum is 169.4 cm and the maximum is 188.7 cm. These two basic statistics give a clear idea about the size of the smallest and tallest of these 6 adults.

## Mean

The mean, also known as average, is probably the most common statistics. It gives an idea on what is the average value, that is, the central value of the data or in other words the center of gravity. The mean is found by summing all values and dividing this sum by the number of observations (denoted \(n\)):

\[mean = \bar{x} = \frac{\text{sum of all values}}{\text{number of values}} = \frac{1}{n}\sum^{n}_{i = 1} x_i\] Below a visual representation of the mean:

Given our sample of 6 adults presented above, the mean is:

\[\bar{x} = \frac{188.7 + 169.4 + 178.6}{6}\] \[\frac{+ 181.3 + 179 + 173.9}{6}\\ = 178.4833\]

In conclusion, the mean size, that is, the average size of our sample of 6 adults is 178.48 cm (rounded to 2 decimals).

## Median

The median is another measure of location so it also gives an idea about the central tendency of the data. The interpretation of the median is that there are as many observations below as above the median. In other words, 50% of the observations lie below the median, and 50% of the observations lie above the median. Below a visual representation of the median:

The easiest way to compute the median is by first sorting the data from lowest to highest (i.e., in ascending order) then take the middle point as the median. From the sorted values, for an odd number of observations, the middle point is easy to find: it is the value with as many observations below as above. Still from the sorted values, for an even number of observations, the middle point is exactly between the two middle values. Formally, after sorting, the median is:

- if \(n\) (number of observations) is odd: \[med(x) = x_{\frac{n+1}{2}}\]
- if \(n\) is even: \[med(x) = \frac{1}{2}\big(x_{\frac{n}{2}} + x_{\frac{n}{2} + 1}\big)\]

where the subscript of \(x\) denotes the numbering of the sorted data. The formulas look harder than they really are, so let’s see with two concrete examples.

### Odd number of observations

Given the height of a sample of 7 adults taken from the 100 adults presented in the introduction:

*188.9*, *163.9*, *166.4*, *163.7*, *160.4*, *175.8* and *181.5*

We first sort the order from lowest to highest:

*160.4*, *163.7*, *163.9*, *166.4*, *175.8*, *181.5* and *188.9*

Given that the number of observations \(n\) is odd (since \(n = 7\)), the median is

\[med(x) = x_\frac{7 + 1}{2} = x_4\]

So we take the fourth value from the sorted values, which corresponds to 166.4. In conclusion, the median size of these 7 adults is 166.4 cm. As you can see, there are 3 observations below 166.4 and 3 observations above 166.4 cm.

### Even number of observations

Now let’s see when the number of observations is even, which is slightly more complicated than when the number of observations is odd. Given the height of a sample of 6 adults:

*188.7*, *169.4*, *178.6*, *181.3*, *179* and *173.9*

We sort the values in ascending order:

*169.4*, *173.9*, *178.6*, *179*, *181.3* and *188.7*

Given that the number of observations \(n\) is even (since \(n = 6\)), the median is

\[med(x) = \frac{1}{2}\big(x_{\frac{6}{2}} + x_{\frac{6}{2} + 1}\big) = \frac{1}{2}\big(x_{3} + x_{4}\big)\]

So we sum the third and fourth values from the sorted values and divide this sum by 2 (which is equivalent than taking the mean of these two middle values):

\[\frac{1}{2}(178.6 + 179) = 178.8\]

In conclusion, the median size of these 6 adults is 178.8 cm. Again, remark that there are as many observations below as above 178.8 cm.

## Mean vs. median

Although the mean and median are often relatively close to each other they should not be confused since they both have advantages and disadvantages in different contexts. Besides the fact that almost everyone knows (or at least have heard about) the mean, it has the advantage that it gives a unique picture for each different series of data. However, it has the disadvantage that the mean is sensible to outliers (i.e., extreme values). On the other hand, the advantage of the median is that it is resistant to outliers and the inconvenient is that it may be the exact same value for very different series of data (so not unique to the data).

To illustrate the “sensible to outlier” argument, consider 3 friends in a bar comparing their salaries. Their salaries are *1800*, *2000* and *2100*€, for an average (mean) salary of *1967*€. A friend of them (who happens to be friend with Bill Gates as well) joins them in the bar. Their salaries are now *1800*, *2000*, *2100* and *1000000*€. The average salary of the 4 friends is now *251475*€, compared to *1967*€ without the rich friend. Although it is statistically correct to say that the average salary of the 4 friends is *251475*€, you will concede that this measure does not represent a fair image of the salaries of the 4 friends, as 3 of them earn much less than the average salary. As we have just seen, the mean is sensible to outliers. On the other hand, if we report the medians, we see that the median salary of the 3 first friends is *2000*€, and the median salary of the 4 friends is *2050*€. As you can see with this example, the median is not sensible to outliers and for series with such extreme value(s), the median is more appropriate compared to the mean as it often gives a better representation of the data. (*Note:* this example also shows how a large majority of people earn less than the average salary reported in the news. This is however beyond the scope of the article.)

Given the previous example, one may then choose to always use the median instead of the mean. However, the median has it own inconvenient which the mean does not have: the median is less unique and less specific to its underlying data than the mean. Consider the following data, representing the grades of 5 students taking a statistics and economics exam:

studentID | economics | statistics |
---|---|---|

1 | 10 | 5 |

2 | 10 | 7 |

3 | 10 | 10 |

4 | 18 | 10 |

5 | 20 | 11 |

The median of the grades is the same in economics and statistics (median = *10*). Therefore, had we computed only the medians, we could have concluded that the students performed as well in economics as in statistics. However, although the medians are exactly the same for both classes, it is clear that students performed better in economics than in statistics. In fact, the mean of the grades in economics is *13.6* and the mean of the grades in statistics is *8.6*. What we have just shown here is that the median is based only on one single value, the middle value, or on the two middle values if there are an even number of observations, while the mean is based on all values (and thus includes more information). The median is therefore not sensible to outliers, but it is also not unique (i.e., not specific) to different series of data, whereas the mean is much more likely to be different and unique for different series of data. This difference in terms of specificity and uniqueness between the two measures may make the mean more useful for data with no outlier.

In conclusion, depending on the context and the data, it is often more interesting to report the mean or the median, or both. As a last remark regarding the comparison between the two most important location measures, note that when the mean and median are equal, the distribution of your data can often be considered to follow a normal distribution (also referred as Gaussian distribution).

## \(1^{st}\) and \(3^{rd}\) quartiles

The first and third quartiles are similar to the median in the sense that they also divide the observations into two parts, except that these parts are not equal. Remind that the median divides the data into two equal parts (with 50% of the observations below and 50% above the median). The first quartile cuts the observations such that there are 25% of the observations **below** and thus 75% **above** the first quartile. The third quartile, as you have guessed by now, represents the value with 75% of the observations below it and thus 25% of the observations above it. There exists several methods to compute the first and third quartile (which sometimes give slight differences, R for instance uses a different method), but here is I believe the easiest one when computing these statistics by hand:

- sort the data in ascending order
- compute \(0.25 \cdot n\) and \(0.75 \cdot n\) (i.e., 0.25 and 0.75 times the number of observations)
- round up these two numbers to the next whole number
- these two numbers represent the rank of the first and third quartile (in the sorted series), respectively

The steps are the same for both an odd and even number of observations. Here is an example with the following series, representing the height in cm of 9 adults:

*170.2*, *181.5*, *188.9*, *163.9*, *166.4*, *163.7*, *160.4*, *175.8* and *181.5*

We first order from lowest to highest:

*160.4*, *163.7*, *163.9*, *166.4*, *170.2*, *175.8*, *181.5*, *181.5* and *188.9*

There are 9 observations so \[0.25 \cdot 9 = 2.25\] and \[0.75 \cdot 9 = 6.75\]

Rounding up to the whole number gives 3 and 7, which represent the rank of the first and third quartiles, respectively. Therefore, the first quartile is 163.9 cm and the third quartile is 181.5 cm.

In conclusion, 25% of adults are less than 163.9 cm tall (and thus 75% of them are more than 163.9 cm tall), while 75% of adults are less than 181.5 cm tall (and thus 25% of them are more than 181.5 cm tall).

### \(q_{0.25}\), \(q_{0.75}\) and \(q_{0.5}\)

Note that the first quartile is denoted \(q_{0.25}\) and the third quartile is denoted \(q_{0.75}\) (where \(q\) stands for quartile). As you can see, the median is actually the second quartile and for this reason it is also sometimes denoted \(q_{0.5}\).

### A note on deciles and percentiles

Deciles and percentiles are similar to quartiles except that they cuts the data in 10 and 100 equal parts. For instance, the \(4^{th}\) decile (\(q_{0.4}\)) is the value such that there are 40% of the observations below it and thus 60% of the observations above it. Furthermore, the \(98^{th}\) percentile (\(q_{0.98}\), also sometimes denoted \(P98\)) is the value such that there are 98% of the observations below it and thus 2% of the observations above it. Percentiles are often used for the weight and height of babies, giving precise information to the parents about where their child stands compared to other children of the same age.

## Mode

The mode of a series is the value that appears most often. In other words, it is the value that has the highest number of occurrences.

Unlike some descriptive statistics that can only be computed for quantitative variables (the mean for instance), the mode can be computed for quantitative **and** qualitative variables (see a recap of the different types of variables if you do not remember the difference).

### Quantitative variables

Given the height of 9 adults:

*170*, *168*, *171*, *170*, *182*, *165*, *170*, *189* and *167*

The mode is 170 because it is the most common value with 3 occurrences. All other values appear only once.

Note that it is possible that a series has no mode (e.g., *4*, *7*, *2* and *10*) or more than one mode (e.g., *4*, *2*, *2*, *8*, *11* and *11*).

Data with two modes are often called bimodal and data with more than two modes are often called multimodal, as opposed to series with one mode which are referred as unimodal.

### Qualitative variables

Given the eye color of the 9 adults presented above:

*brown*, *brown*, *brown*, *brown*, *blue*, *blue*, *blue*, *brown* and *green*

Brown is the most frequent color, so the mode is brown.

# Dispersion

All previous descriptive statistics helps to get a sense of the location and position of the data. We now present the most common dispersion measures, which help to get a sense of the dispersion and the variability of the data (to which extent a distribution is squeezed or stretched):

- range
- standard deviation
- variance
- interquartile range
- coefficient of variation

As for location measures, we detail and compute by hand each of these statistics one by one.

## Range

The range is the difference between the maximum and the minimum value:

\[range = max - min\]

Given the height (in cm) of our sample of 6 adults:

*188.7*, *169.4*, *178.6*, *181.3*, *179* and *173.9*

The range is 188.7 \(-\) 169.4 \(=\) 19.3 cm. The advantage of the range is that it is extremely easy to compute it and it gives a precise idea on what are the possible values in the data. The disadvantage is that it relies on the two most extreme values only.

## Standard deviation

The standard deviation is the most common dispersion measure in statistics. Like the mean for the location measures, if we have to present one statistics which summarizes the spread of the data, it is usually the standard deviation. As its name suggests, the standard deviation tells what is the “normal” deviation of the data. It actually computes the average deviation from the **mean**. The larger the standard deviation, the more scattered the data are. On the contrary, the smaller the standard deviation, the more the data are centred around the mean. Below a visual representation of the standard deviation:

The standard deviation is a bit more complex than the previous statistics in the sense that there are two formulas depending on whether we face a sample or a population. A population includes all members from a specified group, all possible outcomes or measurements that are of interest. A sample consists of some observations drawn from the population, so a part or a subset of the population. For instance, the population may be “**all** people living in Belgium” and the sample may be “**some** people living in Belgium”. Read this article on the difference between population and sample if you want to learn more.

### Standard deviation for a population

The standard deviation for a population, denoted \(\sigma\), is:

\[\sigma = \sqrt{\frac{1}{n}\sum^n_{i = 1}(x_i - \mu)^2}\]

As you can see from the formula, the standard deviation is actually the average deviation of the data from their mean \(\mu\). Note the square for the difference between the observations and the mean to avoid that negative differences are compensated by positive differences.

For the sake of easiness, imagine a population of only 3 adults (the steps are the same with a large population, the computation is just longer). Below their heights (in cm):

*160.4*, *175.8* and *181.5*

The mean is 172.6 (rounded to 1 decimal). The standard deviation is thus:

\[\sigma = \sqrt{\frac{1}{3} \big[(160.4 - 172.6)^2 \\ + (175.8 - 172.6)^2 \\ + (181.5 - 172.6)^2 \big]}\] \[\sigma = 8.91\]

In conclusion, the standard deviation for the heights of these 3 adults is 8.91 cm. This means that, on average, the height of the adults in this population deviates from the mean by 8.91 cm.

### Standard deviation for a sample

The standard deviation for a sample is similar to the standard deviation for a population except that we divide by \(n -1\) instead of \(n\) and it is denoted \(s\):

\[s = \sqrt{\frac{1}{n-1}\sum^n_{i = 1}(x_i - \bar{x})^2}\]

Imagine now that the 3 adults presented in the previous section is a sample instead of a population:

*160.4*, *175.8* and *181.5*

The mean is still 172.6 (rounded to 1 decimal) since the mean is the same whether it is a population or a sample. The standard deviation is now:

\[s = \sqrt{\frac{1}{3 - 1} \big[(160.4 - 172.6)^2 \\ + (175.8 - 172.6)^2 \\ + (181.5 - 172.6)^2 \big]}\] \[s = 10.92\]

In conclusion, the standard deviation for the heights of these 3 adults is 10.92 cm. The interpretation is the same than for a population.

## Variance

The variance is simply the square of the standard deviation. Put it another way, the standard deviation is the square root of the variance. We also distinguish between the variance for a population and for a sample.

### Variance for a population

The variance for a population, denoted \(\sigma^2\), is:

\[\sigma^2 = \frac{1}{n}\sum^n_{i = 1}(x_i - \mu)^2\]

As you can see, the formula for variance is the same than for standard deviation, except that the square root is removed for the variance. Remember the heights of our population of 3 adults:

*160.4*, *175.8* and *181.5*

The standard deviation was 8.91 cm, so the variance of the height of these adults is \(8.91^2 = 79.39\) \(cm^2\) (see below why the unit of a variance is \(unit^2\)).

If you did not know the standard deviation of the population and needed to compute the variance of the population by hand, here is how:

\[\sigma^2 = \frac{1}{3} \big[(160.4 - 172.6)^2 \\ + (175.8 - 172.6)^2 \\ + (181.5 - 172.6)^2 \big] \\ = 79.43\]

(The difference with the above result is due to rounding.)

### Variance for a sample

Again, the variance for a sample is similar to the variance for a population except that we divide by \(n - 1\) instead of \(n\) and it is denoted \(s^2\):

\[s^2 = \frac{1}{n-1}\sum^n_{i = 1}(x_i - \bar{x})^2\]

Imagine again that the 3 adults in the previous section is a sample instead of a population:

*160.4*, *175.8* and *181.5*

The standard deviation for this sample was 10.92 cm, so the variance of the height of these adults is \(10.92^2 = 119.25\) \(cm^2\).

If you did not know the standard deviation of the sample and needed to compute the variance of the sample by hand, here is how:

\[s^2 = \frac{1}{3 - 1} \big[(160.4 - 172.6)^2 \\ + (175.8 - 172.6)^2 \\ + (181.5 - 172.6)^2 \big] \\ = 119.15\]

(The difference with the above result is due to rounding.)

## Standard deviation vs. variance

Standard deviation and variance are often used interchangeably and both quantify the spread of a given dataset by measuring how far the observations are from their mean. However, the standard deviation can be more easily interpreted because the unit for the standard deviation is the same than the unit of measurement of the data (while it is the \(unit^2\) for the variance). Following our example of adult heights in cm, the standard deviation is measured in cm while the variance is measured in \(cm^2\). The fact that the standard deviation keeps the same unit than the initial unit of measurement makes it more interpretable and thus often more used in practice.

### Notations

For completeness, below a table showing the different notations for variance and standard deviation in case of population and sample:

Population | Sample | |
---|---|---|

Standard deviation | \(\sigma\) | \(s\) |

Variance | \(\sigma^2\) | \(s^2\) |

## Interquartile range

Remember the first \(q_{0.25}\) and third quartile \(q_{0.75}\) presented earlier (see this section). The interquartile range is another measure of dispersion of the data, using the quartiles. It is the difference between the third and first quartile:

\[IQR = q_{0.75} - q_{0.25}\]

Considering the height of the 9 adults presented in the section about the first and third quartile:

*170.2*, *181.5*, *188.9*, *163.9*, *166.4*, *163.7*, *160.4*, *175.8* and *181.5*

The first quartile was 163.9 cm and the third quartile was 181.5 cm. The IQR is thus:

\[IQR = 181.5 - 163.9 = 17.6\]

In conclusion, the interquartile range is 17.6 cm. The interquartile range is actually the range (since it is the difference between a higher and a lower value) of the middle data. The graph below may help to understand better the IQR and the quartiles:

## Coefficient of variation

The last dispersion measure is the coefficient of variation. The coefficient of variation, denoted \(CV\), is the standard deviation divided by the mean. Formally:

\[CV = \frac{s}{\bar{x}}\]

Consider the height of a sample of 4 adults:

*163.7*, *160.4*, *175.8* and *181.5*

The mean is \(\bar{x} =\) 170.35 cm and the standard deviation is \(s =\) 9.95 cm. (Find the same values as an exercise!) The coefficient of variation is

\[CV = \frac{9.95 \text{ cm}}{170.35 \text{ cm}} = 0.058 = 5.8\%\]

In conclusion, the coefficient of variation is 5.8%. Note that, as a rule of thumb, a coefficient of variation greater than 15% usually means that the data are heterogeneous while a coefficient of variation equal to or less than 15% means that the data are homogeneous. Given that the coefficient of variation equals 5.8% in this case, we can conclude that these 4 adults are homogeneous in terms of height.

## Coefficient of variation vs. standard deviation

Although the coefficient of variation is rather unknown to the public, it is, in fact, worth presenting when making descriptive statistics.

The standard deviation should always be understood in the context of the mean of the data and is dependent on its unit. The standard deviation has the advantage that it tells by how far on average the data is from the mean in terms of unit in which the data has been measured. Standard deviation is useful when considering variables with same units and approximately same means. However, standard deviation becomes less useful when comparing variables with different units or widely different means. For instance, a variable with a standard deviation of 10 cm cannot be compared to a variable with a standard deviation of 10€ to conclude which one is the most dispersed.

The coefficient of variation is a ratio of two statistics with the same units. It has thus no unit and is independent of the unit in which the data has been measured. Being unit-free, coefficients of variation computed on datasets or variables with different units or widely different means can be compared to conclude, in fine, which data or variables is more (or less) dispersed. For instance, consider a sample of 10 women with their heights in cm and their salaries in €. The coefficients of variation are 0.032 and 0.061 respectively for the height and the salary. We can conclude that, relative to their respective average, their salaries vary more than their heights for these women (since the coefficient of variation is larger for the salary compared to the coefficient variation for the height).

This concludes a relatively long article, thanks for reading! I hope the article helped you to understand and compute the different descriptive statistics by hand. If you would like to learn how to compute these measures in R, read the article “Descriptive statistics in R”.

As always, if you have a question or a suggestion related to the topic covered in this article, please add it as a comment so other readers can benefit from the discussion.

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