## Grouped Data

So far, we’ve worked with data sets that list individual observations. This simply means that for each value, you can see **each individual** observation. For example, take the table below.

**Observation** | **Frequency** |

60 | 1 |

61 | 3 |

62 | 2 |

63 | 5 |

64 | 7 |

65 | 2 |

66 | 1 |

Written out, **this data** would look like the following,

60, 61, 61, 61, 62, 62, 63, 63, 63, 63, 63, 64, 64, 64, 64, 64, 64, 64, 65, 65, 66

Here, finding the mean, median and mode is easy.

**Measure** | **Calculation** |

**Mean** | |

**Median** | The midpoint of the data is 63 |

**Mode** | The most occurring value in the data set is 64 |

However, data doesn’t always come packaged with the individual observations listed. In addition, sometimes you’re not necessarily interested in **understanding** individual data but rather groups within the data.

Grouped data is simply when observations are placed into groups, normally into intervals of some sort. Examples of grouped data include age groups, height groups, time groups and more. While categorical data can also be grouped, for example the frequency of each colour group in a paint store, grouped measures of central tendency make more **intuitive sense** when using only quantitative variables.

In the table below, you’ll find an example of grouped data for age groups.

**Age Groups** | **Frequency** |

0 - 10 | 40 |

10 - 20 | 53 |

20 - 30 | 58 |

30 - 40 | 64 |

40 - 50 | 72 |

50 - 60 | 49 |

60 - 70 | 36 |

70 - 80 | 25 |

Keep in mind that in each group, only the first value, or the lower limit, of the interval is included up until the value before the upper limit, or the last value, of the group. Meaning, while 0 is **included in** the interval 0 through 10, 10 is not included. Instead, 9 is included, meaning there are 10 values in each group because we start counting at 0. Take a look at the image below for clarification.

## Grouped Mean

Finding the **grouped mean** is easy. Simply, follow the formula below.

The table below contains the **explanation** of the notation.

**Element** | **Description** |

| Group mean |

| The frequency of the observation |

| The midpoint of the x |

| The sample size |

Using the example from above, we get the group mean performing the following steps.

**Age Groups** | **Frequency**
| | ** * ** |

**0 - 10** | 40 | | |

**10 - 20** | 53 | 14.5 | 768.5 |

**20 - 30** | 58 | 24.5 | 1421 |

**30 - 40** | 64 | 34.5 | 2208 |

**40 - 50** | 72 | 44.5 | 3204 |

**50 - 60** | 49 | 54.5 | 2670.5 |

**60 - 70** | 36 | 64.5 | 2322 |

**70 - 80** | 25 | 74.5 | 1862.5 |

**Total** | **397** | | **14636.5** |

Plugging this into the formula, we get,

We attain **36.9**, meaning that the mean is somewhere between 30 and 40.

Similarly, finding the median for grouped data requires a different process. To find the group mean, you must follow the **formula** below.

The table below contains the explanation of the notation.

**Element** | **Description** |

| Group median |

| The lower limit of the median group |

| The sample size |

| The cumulative frequency of all groups below the median group |

| The frequency of the group with the median |

| The width of the groups |

Using the same example, we can see that the median of all the groups is roughly the **middle point** of the total frequency.

The 199th point occurs somewhere in the group 30 - 40 (in reality 30 to 39). We can do this estimation because the data are in order.

The **cumulative frequency** can be found in the table below.

**Age Groups** | **Frequency**
| |

**0 - 10** | 40 | |

**10 - 20** | 53 | |

**20 - 30** | 58 | |

**30 - 40** | 64 | 215 |

**40 - 50** | 72 | 287 |

**50 - 60** | 49 | 336 |

**60 - 70** | 36 | 372 |

**70 - 80** | 25 | 397 |

From the previous calculations, we get the following values.

Plugging these values into the formula, we get

Our estimate of the median is about **37**.

## Grouped Mode

The formula for the mode of grouped data is as follows.

The table below contains the **explanation** of the notation.

**Element** | **Description** |

| Group mode |

| The lower limit of the group with the mode (the group with the highest frequency) |

| Frequency of the group with the mode |

| Frequency of the group before the one with the mode |

| Frequency of the group after the one with the mode |

| The width of the groups |

Using the same example, we get the following.

Plugging this into the formula, we get

Which gives us a mode of about **43**.

## Practice Problem 1

Find the group mean of the following data.

**Scores** | **Frequency** |

**1-20** | 5 |

**21 - 40** | 20 |

**41 - 60** | 47 |

**61 - 80** | 15 |

**81 - 100** | 3 |

## Practice Problem 2

Find the group mode of the following data.

**Students** | **Frequency** |

**1-3** | 135 |

**4-6** | 457 |

**7-9** | 549 |

**10-12** | 392 |

## Solution Problem 1

Follow the steps below to find the solution.

The group mean is between 41 and 60.

## Solution Problem 2

The **estimation** of the mode can be found by following the steps below.

Meaning the **estimation** of the mode is 8.1.

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