Skip to content### 4.3 Measures of central tendency and dispersion

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**Content-specific conceptual understandings**

**Measures of central tendency (mean, median, and mode). Estimation of the mean from grouped data.**

Calculation of mean using formula and technology.

Students should use mid-interval values to estimate the mean of grouped data.

**Modal class.**

For equal class intervals only.

**Measures of dispersion (interquartile range, standard deviation, and variance).**

Calculation of standard deviation and variance of the sample using only technology, however, hand calculations may enhance understanding. Variance is the square of the standard deviation.

**Effect of constant changes on the original data.**

Examples: If three is subtracted from the data items, then the mean is decreased by three, but the standard deviation is unchanged.

If all the data items are doubled, the mean is doubled and the standard deviation is also doubled.

**Quartiles of discrete data.**

Using technology. Awareness that different methods for finding quartiles exist and therefore the values obtained using technology and by hand may differ.

**Exercises**

**The mode, median and mean**

- A data set contains these values

2, 3, 3, 3, 6, 6, 7

a) Write down the values of the mode and median.

b) Suppose that a new data value a is added to the set. Find the value of a that would make the mean and the median of the new data set the same. Hence, state the effect of this new data value on the mode of the set. - Find the mode of the data in each frequency table

- The continuous variable X has a frequency distribution as shown in the table.
Find the:

a) mode b) median c) mean. - The depths of snow at a ski resort are measured on 31st January every year for 12 years. All data is in centimetres.

30, 75, 125, 55, 60, 75, 65, 65, 45, 120, 70, 110

Find the range, median, lower quartile, upper quartile, interquartile range. - The graph shows the time that students listen to music during school. Estimate

a) the median time that students listen to music

b) the interquartile range

c) the time a student must spend listening to music to be in the top 10% of listening times. - 50 batteries were tested to see how long they lasted. The results (in hours) are shown in the table. Draw a cumulative frequency diagram and find the median and interquartile range.