Part A: Central Tendency for Ungrouped Data

Measures of central tendency (Mean, Median, and Mode) provide a single value that represents the center of a data set.

Problem 1: Livestock Management

A pastoralist in Puntland records the daily milk yield (in liters) from a specific camel over a week: 8, 10, 7, 12, 10, 9, 14.

  1. Calculate the Mean daily milk yield.
  2. Identify the Mode of the data.
  3. Determine the Median yield.

Part B: Central Tendency for Grouped Data

When data is organized into classes, we use formulas to estimate the central values.

Problem 2: Mean Weight of Infants

A local health clinic recorded the weights of 20 infants.

Weight Range (kg) Frequency (\(f\)) Class Midpoint (\(x\)) \(f \times x\)
2 – 4 4 3 12
4 – 6 8 5 40
6 – 8 5 7 35
8 – 10 3 9 27

Task: Calculate the estimated Mean weight.

Problem 3: Mode for Grouped Data

Find the Mode for the following distribution of customers visiting a shop:

Number of Customers Frequency (\(f\))
10 – 20 5
20 – 30 12
30 – 40 8
40 – 50 5

Problem 4: Median for Grouped Data

The heights of 40 students were recorded. Calculate the Median height.

Height (cm) Frequency (\(f\)) Cumulative Frequency (\(C_f\))
140 – 150 6 6
150 – 160 14 20
160 – 170 15 35
170 – 180 5 40

Part C: Correlation

Correlation describes the strength and direction of a relationship between two variables.

Problem 5: Educational Study

A teacher plots the relationship between hours spent studying (\(x\)) and test scores (\(y\)).

[Image of a scatter plot showing positive correlation]

  1. If the points trend upward from left to right, what type of correlation is present?
  2. If the points are scattered randomly with no trend, what is the correlation called?

Part D: Mode and Median for Grouped Data

When data is organized into classes (grouped data), we use specific formulas to estimate the median and mode since we no longer have the individual raw data points.

1. The Mode of Grouped Data

The mode for grouped data is estimated using the modal class (the class with the highest frequency).

Formula: \[\text{Mode} = L + \left( \frac{f_1 - f_0}{(f_1 - f_0) + (f_1 - f_2)} \right) \times c\]

  • \(L\): Lower class boundary of the modal class.
  • \(f_1\): Frequency of the modal class.
  • \(f_0\): Frequency of the class preceding the modal class.
  • \(f_2\): Frequency of the class following the modal class.
  • \(c\): Class width.

Problem 5: Market Research A shopkeeper records the daily number of customers over a month. Find the Mode for the following distribution:

Number of Customers Frequency (\(f\))
10 – 20 5
20 – 30 12
30 – 40 8
40 – 50 5

2. The Median of Grouped Data

The median is the value that falls at the \(\frac{N}{2}\) position of the total frequency.

Formula: \[\text{Median} = L + \left( \frac{\frac{N}{2} - C_f}{f} \right) \times c\]

  • \(L\): Lower class boundary of the median class.
  • \(N\): Total frequency (\(\sum f\)).
  • \(C_f\): Cumulative frequency of the class before the median class.
  • \(f\): Frequency of the median class.
  • \(c\): Class width.

Problem 6: Student Heights The heights of 40 students in a Form Three class were recorded. Calculate the Median height.

Height (cm) Frequency (\(f\)) Cumulative Frequency (\(C_f\))
140 – 150 6 6
150 – 160 14 20
160 – 170 15 35
170 – 180 5 40