krit.club logo

Data Handling - Mean, Median, Mode, and Range

Grade 5IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

•

The Mean (Average) represents the balance point of a data set. Imagine several towers of blocks with different heights; if you were to move blocks from the taller towers to the shorter ones until every tower was the same height, that height would be the Mean. It is calculated by dividing the total sum of values by the number of values in the set.

•

The Median is the middle value in a set of data that has been arranged in order from least to greatest. Think of it like the 'median' strip in the middle of a highway, which splits the road into two equal sides. If there is an odd number of values, the median is the exact center number. If there is an even number of values, the median is the average of the two middle numbers.

•

The Mode is the value that appears most frequently in a data set. In a bar graph, the mode is easily identified as the tallest bar. A data set can have one mode, more than one mode (if multiple values appear the same highest number of times), or no mode at all if every value appears only once.

•

The Range measures the spread of the data by finding the difference between the largest and smallest values. On a number line, the range is the total distance between the point furthest to the left and the point furthest to the right. A large range means the data is widely spread out, while a small range means the data points are close together.

•

Frequency describes how many times a specific value occurs within a data set. This is often recorded using a tally chart, where each vertical mark represents one occurrence, and a diagonal mark across four vertical lines represents a group of five. Frequency is what determines the height of bars in a histogram or bar chart.

•

Outliers are data points that are significantly higher or lower than the rest of the values in a set. Visually, an outlier looks like a lone dot far away from a cluster of other dots on a dot plot. Outliers can greatly affect the mean by pulling it up or down, but they usually have very little impact on the median.

📐Formulae

Mean=Sum of all valuesTotal number of values\text{Mean} = \frac{\text{Sum of all values}}{\text{Total number of values}}

Range=Maximum value−Minimum value\text{Range} = \text{Maximum value} - \text{Minimum value}

Median (Odd number of values)=Value at position n+12\text{Median (Odd number of values)} = \text{Value at position } \frac{n + 1}{2}

Median (Even number of values)=Sum of two middle values2\text{Median (Even number of values)} = \frac{\text{Sum of two middle values}}{2}

💡Examples

Problem 1:

Find the Mean, Median, Mode, and Range for the following test scores: 85,90,75,90,10085, 90, 75, 90, 100.

Solution:

  1. Mean: Sum =85+90+75+90+100=440= 85 + 90 + 75 + 90 + 100 = 440. There are 55 scores. Mean =4405=88= \frac{440}{5} = 88.
  2. Median: Arrange in order: 75,85,90,90,10075, 85, 90, 90, 100. The middle value is the 3rd number, which is 9090.
  3. Mode: The number 9090 appears twice, while others appear once. Mode =90= 90.
  4. Range: High =100= 100, Low =75= 75. Range =100−75=25= 100 - 75 = 25.

Explanation:

To solve this, first total the scores for the mean, then sort them to identify the middle (median) and the most frequent (mode) values. Finally, subtract the lowest from the highest for the range.

Problem 2:

Calculate the Median and Range for this data set: 12,5,8,3,10,1212, 5, 8, 3, 10, 12.

Solution:

  1. Order the data: 3,5,8,10,12,123, 5, 8, 10, 12, 12.
  2. Median: There are 66 values (even). The middle two are 88 and 1010. Median =8+102=182=9= \frac{8 + 10}{2} = \frac{18}{2} = 9.
  3. Range: Maximum is 1212 and Minimum is 33. Range =12−3=9= 12 - 3 = 9.

Explanation:

Since there is an even number of data points (n=6n=6), the median is the average of the 3rd and 4th values after sorting. The range shows the spread between the lowest and highest values.