Central tendency describes the center or typical value of a dataset. The three main measures are mean, median, and mode.

2.1 Mean (Arithmetic Average)

The mean is the sum of all values divided by the total number of values.

For ungrouped data: $\text{Mean} = \frac{\sum x}{N}$

Where Σx is the sum of all values and N is the total number of values.

Example 3: Find the mean of these daily temperatures (°C) recorded in Arusha: 22, 24, 19, 21, 25

$\text{Mean} = \frac{22 + 24 + 19 + 21 + 25}{5} = \frac{111}{5} = 22.2°C$

For grouped data (using class marks): $\text{Mean} = \frac{\sum (f \times x)}{\sum f}$

Where f is frequency and x is the class mark.

Example 4: Calculate the mean profit from the grouped data above.

$\text{Mean} = \frac{1510}{40} = 37,750 \text{ Tsh}$

2.2 Median

The median is the middle value when data is arranged in order. For an odd number of values, it's the middle one. For an even number, it's the average of the two middle values.

Example 5: Heights of 7 students (cm): 145, 152, 148, 155, 150, 147, 149

Arrange in order: 145, 147, 148, 149, 150, 152, 155

The 4th value (149 cm) is the median.

Example 6: Heights of 6 students (cm): 145, 152, 148, 155, 150, 147

Arrange: 145, 147, 148, 150, 152, 155

Median = average of 3rd and 4th values = (148 + 150) ÷ 2 = 149 cm

2.3 Mode

The mode is the value that occurs most frequently.

Example 7: Number of cups of tea sold daily at a cafeteria: 2, 3, 4, 3, 3, 2, 5, 3, 3

The mode is 3 (it appears 5 times).

A histogram is a graphical display of data using bars of different heights. Unlike bar charts, histograms have no gaps between bars because the data is continuous (grouped into intervals).

3.1 Drawing a Histogram

Steps:

1. Draw the class intervals on the horizontal axis
2. Draw frequencies on the vertical axis
3. Draw bars with heights equal to the frequencies
4. Use appropriate scales

Example 8: Draw a histogram for the data below showing the distribution of marks obtained by Form 4 students in a Mathematics test:

Solution:

The histogram will have:

• Vertical axis: frequency (scale: 1 cm = 5 students)
• Horizontal axis: marks (continuous scale)

The bars will be drawn with heights corresponding to frequencies: 3, 12, 25, 15, and 5 respectively. Since this is continuous data, the bars touch each other.

3.2 Interpreting Histograms

From a histogram, you can:

• Identify the most common class (tallest bar)
• See the spread of data (wide or narrow)
• Observe the shape of distribution (symmetrical, skewed)

A frequency polygon is a line graph that shows the shape of a distribution. It is drawn by joining the midpoints (class marks) of the tops of histogram bars.

4.1 Drawing a Frequency Polygon

Steps:

1. Find the class mark for each interval
2. Plot points with class marks on horizontal axis and frequencies on vertical axis
3. Add one class below the lowest and one above the highest, both with zero frequency
4. Join the points with straight lines

Example 9: Draw a frequency polygon for the test marks data above.

Add extra intervals: Below 0 (class mark -10.5, frequency 0), Above 100 (class mark 110, frequency 0).

Plot the points and join them to form the polygon.

An ogive (cumulative frequency curve) shows how many observations lie below a particular value. It helps us find medians and other percentiles.

5.1 Cumulative Frequency Table

To create a cumulative frequency table, add each frequency to the total of all previous frequencies.

Example 10: Create a cumulative frequency table for the test marks data.

5.2 Drawing an Ogive

Steps:

1. Find the upper class boundaries (real upper limits) for each class
2. Plot cumulative frequency against upper class boundary
3. Join points with a smooth curve
4. Add a point at the lower boundary of the first class with zero frequency

For the data above, using real limits (0.5, 19.5, 39.5, 59.5, 79.5, 100.5):

5.3 Using the Ogive to Find the Median

The median is the value corresponding to half the total frequency (N/2).

From Example 10: N = 60, so N/2 = 30

On the ogive, find 30 on the vertical axis, draw a horizontal line to meet the curve, then drop vertically to read the median on the horizontal axis.

For this data, the median is approximately 50 marks.

Statistics is widely used in Tanzania in everyday situations. For example, when a shopkeeper at Kariakoo Market records daily sales over several months, they can use frequency distributions to see which products sell most often, calculate the average (mean) daily sales to plan stock, and draw an ogive to determine the sales figure that half of their daily earnings exceed (the median). This helps in making informed business decisions about which items to prioritize and how much capital to allocate for restocking.