Which one of the following measures of central tendency will be used to determine the average size of the shoe sold in the shop?
Mode
Understanding the right measure of central tendency is crucial for analyzing different types of data. The question asks for the best measure to determine the average size of shoes sold in a shop. When thinking about the "average size" in a retail context like shoe sales, we are most interested in the size that sells the most frequently, as this helps in managing inventory and understanding customer demand.
Identifying the Best Measure for Average Shoe Size
Let's look at the options provided and how each measure of central tendency is defined and used:
- Arithmetic Mean: This is calculated by summing all the values in a data set and dividing by the number of values. It represents the mathematical average. While shoe sizes are numerical, the arithmetic mean might result in a size that is not an actual shoe size (e.g., average size 7.3). More importantly, it doesn't tell you which size is most popular.
- Geometric Mean: This is used primarily for averaging rates, ratios, or growth rates. It is not appropriate for finding the average shoe size sold.
- Median: This is the middle value in a data set that has been ordered from least to greatest. It divides the data into two equal halves. While the median gives a sense of the center of the data, it doesn't necessarily represent the most frequently occurring value or the most popular size. For example, if a shop sold sizes 6, 7, 7, 8, 10, the median is 7, but if it sold sizes 6, 7, 8, 8, 8, the median is 8, and so is the mode. However, if it sold sizes 6, 7, 7, 8, 10, 10, 10, the median is 8, while the mode is 10 (the most popular).
- Mode: This is the value that appears most frequently in a data set. In the context of shoe sizes sold, the mode tells us which size was sold the highest number of times. This is exactly what a shop owner needs to know to stock the most popular shoe size.
Why Mode is Ideal for Average Shoe Size Sold
When determining the "average size of the shoe sold," the most practical interpretation is finding the size that is most popular among customers. This popularity is directly measured by the frequency with which each size is sold. The measure of central tendency that identifies the most frequent value in a dataset is the Mode.
For instance, consider the following sales data for shoe sizes over a week:
Sizes sold: 6, 7, 7, 8, 8, 8, 8, 9, 10, 10
- Arithmetic Mean: \(\frac{6+7+7+8+8+8+8+9+10+10}{10} = \frac{81}{10} = 8.1\) (Not an actual shoe size, doesn't represent popularity)
- Median: Order the data: 6, 7, 7, 8, 8, 8, 8, 9, 10, 10. The middle values are 8 and 8. Median = 8. (Represents the center, but not the most frequent if distribution was different)
- Mode: The size 8 appears 4 times, which is more than any other size. The mode is 8. (Represents the most popular size)
In this example, the mode (size 8) clearly indicates the size that sold the most. This is the most useful information for a shoe shop owner wanting to determine the "average size" in terms of popularity and stocking needs.
Therefore, the measure of central tendency used to determine the average size of the shoe sold in the shop, interpreted as the most popular size, is the Mode.
Revision Table: Measures of Central Tendency
| Measure | Definition | Best Use Case Example | Suitability for Average Shoe Size Sold |
|---|---|---|---|
| Arithmetic Mean | Sum of values divided by number of values. | Average height of students in a class. | Not ideal; doesn't show most frequent value, result might not be actual size. |
| Geometric Mean | N-th root of the product of N values. | Average growth rate over multiple periods. | Not suitable. |
| Median | Middle value in an ordered dataset. | Median income of a neighborhood (less affected by outliers). | Less suitable than Mode; doesn't guarantee representation of the most frequent value. |
| Mode | The most frequently occurring value. | Most popular car color sold. | Ideal; directly indicates the most frequently sold (most popular) size. |
Additional Information on Central Tendency
Measures of central tendency are single values that attempt to describe a set of data by identifying the central position within that set. They are also known as measures of central location. Choosing the appropriate measure depends heavily on the type of data and what you want to represent as the 'average'.
- For nominal data (categories like colors, types of cars), only the Mode can be used.
- For ordinal data (ranked data like survey responses: agree, strongly agree), the Median is often used, along with the Mode.
- For interval or ratio data (numerical data like height, weight, sales figures), Mean, Median, and Mode can all be used, but the choice depends on the distribution (e.g., Median is better if data is skewed).
In the case of shoe sizes, although they are numerical, the context of retail sales prioritizes popularity (frequency) over a mathematical average or central position, making the Mode the most relevant measure of central tendency.
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Important Questions from Measures of Central Tendency
A random sample of 20 people is classified in the following table according to their ages:
Age
Frequency
15 – 25
2
25 – 35
4
35 – 45
6
45 – 55
5
55 - 65
3
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If the difference of mode and median is 36, then the difference of median and mean is:
In a Mathematics test 15 students scored 80 marks, 20 students scored 75 marks, 28 students scored 65 marks and 25 students scored 60 marks, mode of the score is:
If the mode of the scores 10, 12, 13, 15, 15, 13, 12, 10, x is 15, then what is the value of x?
