Presentation of Data: Frequency Distributions

Presentation of data involves organizing and summarizing raw data to make it easier to understand and analyze. This includes techniques to present raw data in a structured manner to reveal patterns and insights. We will discuss two main ways: frequency distributions.

Raw Data: This is the original, unorganized data collected from a survey, experiment, or observation. It’s the data as it’s first recorded, before any processing or summarization.

Frequency Distribution: This is a table or graph that shows how often each value (or a range of values) appears in a dataset. There are two main types:

• Ungrouped Frequency Distribution: Each individual value in the dataset is listed along with its frequency (the number of times it occurs). This is suitable for datasets with a small number of unique values.
• Grouped Frequency Distribution: Data is organized into class intervals (ranges). Each class interval is listed along with its frequency. This is useful for large datasets with a wide range of values. The choice of class intervals impacts the presentation and can influence how patterns are seen.

Formulae

Relative Frequency: The proportion of the data values that fall into a particular class or have a specific value.

$Relative \ Frequency = \frac{Frequency \ of \ Class \ or \ Value}{Total \ Number \ of \ Observations}$

Cumulative Frequency: The running total of frequencies. In an ungrouped distribution, it’s the sum of all frequencies up to and including a given value. In a grouped distribution, it’s the sum of frequencies up to and including the upper limit of each class interval.

Examples

Example-1: Ungrouped Frequency Distribution

Suppose we surveyed 20 students about the number of siblings they have, with the following raw data: 0, 1, 1, 2, 0, 2, 1, 1, 3, 0, 2, 1, 0, 2, 1, 1, 0, 2, 3, 1

We can create an ungrouped frequency distribution:

Number of Siblings (x)	Frequency (f)	Relative Frequency	Cumulative Frequency
0	5	5/20 = 0.25	5
1	8	8/20 = 0.40	13
2	5	5/20 = 0.25	18
3	2	2/20 = 0.10	20
Total	20	1.00

Example-2: Grouped Frequency Distribution

Suppose we have the following test scores from 30 students: 65, 70, 72, 75, 78, 80, 82, 83, 85, 85, 87, 88, 90, 90, 91, 92, 93, 94, 95, 96, 68, 71, 73, 77, 79, 81, 84, 86, 89, 92

Let’s create a grouped frequency distribution using class intervals of size 5:

Class Interval	Frequency (f)	Relative Frequency	Cumulative Frequency
65-69	3	3/30 = 0.10	3
70-74	4	4/30 = 0.13	7
75-79	4	4/30 = 0.13	11
80-84	5	5/30 = 0.17	16
85-89	7	7/30 = 0.23	23
90-94	6	6/30 = 0.20	29
95-99	1	1/30 = 0.03	30
Total	30	1.00

Common mistakes by students

• Incorrect Class Intervals: When creating grouped frequency distributions, students often make errors in defining the class intervals (e.g., overlapping intervals, gaps between intervals).
• Miscounting Frequencies: Students can miscount the number of data points that fall within each class interval or for a specific value.
• Misunderstanding Relative and Cumulative Frequencies: Confusing the calculations and purposes of relative and cumulative frequencies.
• Ignoring the Total: Forgetting to calculate and check the total frequency to ensure it matches the total number of data points.

Real Life Application

• Surveys: Analyzing the results of surveys to understand consumer preferences, opinions, and behaviors.
• Sales Data: Businesses use frequency distributions to understand sales trends, identify popular products, and track sales performance over time.
• Health Statistics: Health organizations use frequency distributions to analyze the distribution of diseases, track mortality rates, and monitor the effectiveness of public health programs.
• Weather Analysis: Meteorologists use frequency distributions to analyze temperature, rainfall, and other weather patterns.

Fun Fact

The concept of frequency distribution is fundamental in statistics and data analysis. The shape of a frequency distribution can reveal a lot about the underlying data. For example, a bell-shaped distribution (normal distribution) is common in many natural phenomena, while skewed distributions indicate that data is not symmetrically distributed.

Recommended YouTube Videos for Deeper Understanding

Q.1 Which of the following is NOT a characteristic of raw data?

Check Solution

Ans: C

Raw data is typically unorganized and requires processing before analysis.

Q.2 The following data represents the scores of 10 students on a math test: 60, 75, 80, 60, 90, 85, 75, 60, 95, 70. What is the frequency of the score 60 in an ungrouped frequency distribution?

Check Solution

Ans: C

The score 60 appears 3 times in the data set.

Q.3 In a grouped frequency distribution, the class interval is 10-19. What is the class width?

Check Solution

Ans: B

Class width is calculated by subtracting the lower limit from the upper limit plus 1: 19-10+1 = 10

Q.4 If the lower limit of a class interval is 20 and the class width is 5, what is the upper limit of the class interval?

Check Solution

Ans: B

The upper limit is found by adding the class width minus 1 to the lower limit: 20 + 5 – 1 = 24

Q.5 Which of the following statements is true about the advantages of using grouped frequency distribution?

Check Solution

Ans: B

Grouped frequency distribution makes large datasets easier to analyze.

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