Contents

# Mode|Definition & Meaning

**Definition**

**Statistics** **defines** the **mode** as the **value** that **occurs** most **frequently** in a **dataset.** In a dataset, it tells us **where most** of the **values** are **concentrated,** which is **known** as the **central tendency.**

**Conceptual OverviewÂ **

Figure 1 – Conceptual Overview of Mode

As a **general rule,** we **represent data** in **statistics** by a **representative value** that **describes** the **whole collection** in a rough sense. The **measure** of **central tendency refers** to **this representative value.** According to its name, it is a **value** that is the **basis for analyzing data.**

In this way, **we can create** a **statistical summary** of a **large amount** of **organized data** by **using** measures of **central tendency.** The **mode** of **data** can be **used** to **measure central tendency.**

For instance, **talking about** a **football match,** suppose there are **5 football matches** and a **player** named **Ronaldo scores** 1 goal in the **first match,**Â **2 goals** in the **second**Â **match,** **1 goal** in the **third match,** and **2 goals** in the **third** and **fourth match.**Â

**We** can **see** the **data** is kind of **rough** and we **can’t build** an **argument** based on this data so what we do is we will **make** a **tabular form** and use the **concept** of **mode** to **build** the **conclusion** or **argument** about the data as illustrated above, we can see the **number** of **frequency** of scoring **two goals** is **greater** than rest of all goal so we can **conclude** that, **Ronaldo scores** **2 goals frequently**.

## Generic Procedure for Finding Mode

Figure 2 -Procedure For Finding Mode

To find the mode of a dataset, you can follow these steps:

- A
**list**or**table**should be**created**with each**data point listed**separately. **Analyze**each data point to**determine**its**frequency.**A dataset’s frequency is the number of times a particular data point appears.**Identify**the**data point**or points that**occur most frequently.**These are the modes of the dataset.

**Mode Vs. Median Vs. Mean**

**Mean**

A dataset’s **mean** is **determined** by **adding together** all of its **values** and **dividing** by the **total number** of **values** in the dataset. When the **data** is **continuous** and there are **no outliers,** the **mean** provides a **helpful indicator** of **central tendency** (extreme values).

**Median**

In a **dataset** that has been **sorted** in **numerical order,** the **median** is the **midway value.** Given that it is **unaffected** by **extreme values,** the **median** is a **good indicator** of central tendency when the data is skewed or contains outliers.

**Mode**

The **value** that **appears most frequently** in a dataset is the mode. When the data is categorical or ordinal, the **mode** is a **valuable indicator** of **central tendency** (meaning it can be ranked or ordered).

**The Formula for Finding Mode**

**Ungrouped DataÂ **

To find the mode **for ungrouped data,** you can use the following formula:

**Mode** = value that occurs most frequently

This means that you simply need to **count** the **number** of **times** each **value appears** in the **dataset,** and choose the value that appears most often.

**Grouped Data**

**Mode** = $\dfrac{\Sigma F \times \text{Midpoint}}{ \Sigma F}$

Where:

**F**is the**frequency**of each group.**Midpoint**is the**midpoint**of**each group,**calculated as:

\[\dfrac{\text{Lower bound of the group} + \text{Width of the group}}{2}\]

**Relationship Between Mean, Median, and Mode**

The **relationship** between the **mode, median,** and **mean** can be summarized as follows:

- The
**dataset**is**symmetrical**if the**mode**and**median**are**identical,**and the**mean**is**identical**to the**mode**and**median**as**well.** - The
**dataset**is**skewed**if the**mode**and**median**are**not**the**same.**The**dataset**is**skewed**to the**right**if the**mode**is**higher**than the median (positively skewed). The**dataset**is**skewed**to the**left**if the**mode**is**less**than the**median**(negatively skewed). In this instance, the mean might or might not be the same as the mode and median.

**Finding Mode Using Mean and MedianÂ **

**Using Mean**

The **result** that is **most similar** to the **dataset’s mean** is the **mode.** The **dataset’s mean** can be **determined first,** and the **value** that is most **similar** to the **mean** can then be **used** to **determine** the **mode** using this formula.

**Using Median**

The mode is the **value** that is **closest** to the **median** of the **dataset.** By **calculating** the **dataset’s median** value and then selecting the value that is most similar to the median, you can use this formula to determine the mode.

**Properties of Mode**

There are a few key properties of the mode that are worth noting:

**Extreme values**do**not have**an**impact**on the**mode**(also known as outliers).- Sometimes the
**mode**is**not particular.**There are**two possible modes**for a**dataset:**many modes and no modes at all. - For
**datasets**that are**skewed**or contain a significant number of outliers, the**mode**is**not always**a**reliable indicator**of central tendency. - For
**datasets**with a lot of**unique values,**the**mode**is**not**always a**good indicator**of central tendency.

Generally speaking, the **mode** is most **helpful** when the **data** is **categorical** or **ordinal** (meaning it can be ranked or ordered). Because it represents the most frequent value or category in the dataset, the **mode** can be a **helpful indicator** of **central tendency** in certain circumstances.

When the **data** is **continuous** or has a **large number** of **distinct values,** the mode may not be as helpful because it may not truly reflect the data’s general distribution.

**Solved Example Problems of Mode**

**Example 1**

**Consider** the following **dataset:** **2, 4, 6, 8, 10, 10, 12, 14, 16, and 18**. **Find** the **mode** of the dataset.

**Solution**

Figure 3 – Example of the mode of ungrouped data

To **find** the **mode** of this dataset, we can **count** the **number** of times **each value appears.**

In this dataset, the data point “**10**” **occurs** most **frequently,** with a **frequency** of 2. While the rest of the data points occurred only one time having a frequency equal to 1. Therefore, the **mode** of this **dataset** is **10.**

**Example 2**

For example, **consider** the **following grouped data** having four groups. The frequency and midpoint of each group are shown in the table below.

Figure 4 – Example of the mode of grouped data

**Solution**

To **find** the **mode** for this **dataset** using the formula, we can follow these steps:

**Calculate**the**midpoint**for each group:**12, 17, 22, and 27.****Multiply**the**frequency**for each group**by**the**midpoint**to obtain the**weighted midpoint:****60, 136, 66, and 54.****Add**up all the**frequencies:****5 + 8 + 3 + 2 = 18.****Add**up all the**weighted midpoints:****60 + 136 + 66 + 54 = 316.****Plug**the**values**into the**formula: mode**= (316 / 18) =**17.6.**- In this example, the
**mode**is**17.6.**This**means**that the**value**that**occurs most frequently**in the**dataset**is**around 17.6.**

When a **dataset contains categorical** or **ordinal data,** the **mode** can be **useful** for **spotting patterns** or trends. Additionally, **finding** the **most frequent value** in a dataset can be **useful** for **making judgments** or **predictions.**

*All mathematical drawings and images were created with GeoGebra.*