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Data Handling - Organizing Data in Frequency Tables

Grade 6IB

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

🔑Concepts

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Raw Data: This refers to the data collected in its original form before any organization. For example, if you record the scores of 20 students in a list like 15,12,15,10...15, 12, 15, 10..., this unorganized list is your raw data.

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Tally Marks: A visual counting system used to record data points as they are counted. Each occurrence is marked by a vertical line ∣|. To make counting easier, every fifth mark is drawn as a diagonal line through the previous four vertical lines, creating a 'bundle' or 'gate' shape that represents exactly 5 units.

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Frequency: The total number of times a particular data value or category occurs in a dataset. In a table, this is recorded as a numeral in the 'Frequency' column, corresponding to the total count of tally marks for that row.

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Frequency Distribution Table: A structured grid used to organize data. It typically contains three columns: the first for the Category or Value (e.g., 'Color' or 'Score'), the second for Tally Marks, and the third for the numerical Frequency. Visually, it transforms a messy list into a clear summary.

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Discrete Data: Data that can only take specific, separate values, often counted in whole numbers. Examples include the number of goals scored in a match or the number of pets in a household. On a table, these are listed as individual values in the first column.

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Grouped Data (Class Intervals): When there is a large range of data, values are grouped into intervals called 'classes' (e.g., 10−1910-19, 20−2920-29). This prevents the frequency table from becoming too long. Each interval must be of equal size and should not overlap.

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Total Frequency: The sum of all individual frequencies in the table, represented by the symbol ∑f\sum f. It should always equal the total number of data points originally collected, serving as a check for accuracy.

📐Formulae

Total Frequency (∑f)=f1+f2+f3+⋯+fn\text{Total Frequency } (\sum f) = f_1 + f_2 + f_3 + \dots + f_n

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

Relative Frequency=Frequency of a categoryTotal Frequency\text{Relative Frequency} = \frac{\text{Frequency of a category}}{\text{Total Frequency}}

💡Examples

Problem 1:

The following are the marks obtained by 15 students in a math mini-quiz: 5,7,5,8,6,7,5,9,8,7,6,5,10,7,85, 7, 5, 8, 6, 7, 5, 9, 8, 7, 6, 5, 10, 7, 8. Organize this data into a frequency table.

Solution:

Step 1: Identify the distinct marks: 5,6,7,8,9,105, 6, 7, 8, 9, 10. Step 2: Use tally marks to count occurrences:

  • 55: ∣∣∣∣|||| (Frequency: 44)
  • 66: ∣∣|| (Frequency: 22)
  • 77: ∣∣∣∣|||| (Frequency: 44)
  • 88: ∣∣∣||| (Frequency: 33)
  • 99: ∣| (Frequency: 11)
  • 1010: ∣| (Frequency: 11) Step 3: Total Frequency ∑f=4+2+4+3+1+1=15\sum f = 4 + 2 + 4 + 3 + 1 + 1 = 15.

Explanation:

We list the unique scores in the first column, record the tallies for each score in the second, and write the final count in the frequency column. Summing the frequencies ensures no data points were missed (1515 matches the original count).

Problem 2:

A researcher records the number of hours 10 students spent studying in a week: 12,15,12,18,20,15,12,18,12,2012, 15, 12, 18, 20, 15, 12, 18, 12, 20. Calculate the range and the relative frequency of students who studied for 1212 hours.

Solution:

Step 1: Find the Range. Maximum=20\text{Maximum} = 20, Minimum=12\text{Minimum} = 12. Range=20−12=8\text{Range} = 20 - 12 = 8. Step 2: Find frequency of 1212 hours. Looking at the data, 1212 appears 44 times. Step 3: Calculate Relative Frequency. Total Frequency=10\text{Total Frequency} = 10. Relative Frequency=410=0.4\text{Relative Frequency} = \frac{4}{10} = 0.4 or 40%40\%.

Explanation:

The range tells us the spread of the data (88 hours). The relative frequency shows the proportion of the total group that falls into a specific category, calculated by dividing the specific frequency by the total number of observations.