Raw, ungrouped & grouped data

Raw Data: Data that has not been organized or processed in any way. It is the original data collected from a source.

Ungrouped Data: Data that has been organized but not grouped into classes or intervals. It is presented as a list of individual values.

Grouped Data: Data that has been organized into groups or classes, often with a corresponding frequency. It is commonly used to

 summarize large datasets.

Raw Data

Raw data is data that has not been organized or processed in any way. It is the original data collected from a source.

Example: Consider the following raw data representing the heights (in inches) of 10 students: 62, 65, 68, 69, 70, 71, 72, 73, 74, 75

Ungrouped Data

Ungrouped data is data that has been organized but not grouped into classes or intervals. It is presented as a list of individual values.

Example: Using the same height data, we can organize it in ascending order: 62, 65, 68, 69, 70, 71, 72, 73, 74, 75

Calculating Measures for Ungrouped Data:

• Mode: The most frequent value. In this case, there is no mode as all values occur once.

• Mean: The sum of all values divided by the number of values.

  • Mean = (62+65+68+69+70+71+72+73+74+75)/10 = 70 inches

• Trimmed Mean: Remove a certain percentage of data from both ends and calculate the mean. For example, a 10% trimmed mean would remove the lowest and highest values:

  • Trimmed Mean = (65+68+69+70+71+72+73+74)/8 = 70.25 inches

• Lower Quartile (Q1): The value below which 25% of the data falls.

  • Q1 = 68 inches

• Median (Q2): The middle value.

  • Median = (70+71)/2 = 70.5 inches

• Upper Quartile (Q3): The value below which 75% of the data falls.

  • Q3 = 73 inches

• Percentiles: The value below which a certain percentage of the data falls. For example, the 90th percentile is the value below which 90% of the data falls.  

• Interquartile Range (IQR): The difference between the third and first quartiles.

  • IQR = Q3 - Q1 = 73 - 68 = 5 inches

• Semi-Interquartile Range: Half of the IQR.

  • Semi-IQR = IQR/2 = 5/2 = 2.5 inches

• Standard Deviation: A measure of how spread out the data is.

  • Standard Deviation = 3.42 inches (calculated using a calculator or statistical software)

• Variance: The square of the standard deviation.

  • Variance = 11.73 square inches

Grouped Data

Grouped data is data organized into classes or intervals with corresponding frequencies.

Example: Consider the following grouped data representing the ages of 50 people:

Calculating Measures for Grouped Data:

• Mode: The modal class is the class with the highest frequency (30-39).

• Mean: Calculated using the midpoint of each class interval and the frequency.

• Median: Determined by finding the cumulative frequency and identifying the class interval containing the median value.