Question

question 1-20
the following data are final exam scores of 10 randomly selected students in a statistics class:
85, 88, 90, 99, 75, 80, 80, 70, 91, 82.
which of the following is the measure of central tendency with the smallest value?
○ mean
○ median
○ mode

Explanation:

Step1: Calculate the Mean

First, sum all the scores: \(85 + 88 + 90 + 99 + 75 + 80 + 80 + 70 + 91 + 82\). Let's compute that:
\(85+88 = 173\); \(173+90 = 263\); \(263+99 = 362\); \(362+75 = 437\); \(437+80 = 517\); \(517+80 = 597\); \(597+70 = 667\); \(667+91 = 758\); \(758+82 = 840\).
The mean is the sum divided by the number of data points (10), so \(\text{Mean} = \frac{840}{10} = 84\).

Step2: Calculate the Median

First, order the data set: \(70, 75, 80, 80, 82, 85, 88, 90, 91, 99\).
Since there are 10 (even) data points, the median is the average of the 5th and 6th values.
The 5th value is 82, the 6th is 85. So \(\text{Median} = \frac{82 + 85}{2} = \frac{167}{2} = 83.5\).

Step3: Calculate the Mode

The mode is the most frequently occurring value. In the data set, 80 appears twice, and all other values appear once. So Mode = 80.

Now, compare the three: Mean = 84, Median = 83.5, Mode = 80. The smallest is the Mode? Wait, no, wait: Wait, 80 (Mode) is 80, Median is 83.5, Mean is 84. Wait, but wait, did I order the data correctly? Let's re - order: 70, 75, 80, 80, 82, 85, 88, 90, 91, 99. Yes. Sum: 70+75 = 145; +80 = 225; +80 = 305; +82 = 387; +85 = 472; +88 = 560; +90 = 650; +91 = 741; +99 = 840. Correct. Mean is 84. Median: (82 + 85)/2 = 83.5. Mode is 80. So the smallest measure of central tendency is the Mode? Wait, but wait, the options are Mean, Median, Mode. Wait, 80 (Mode) is less than 83.5 (Median) and 84 (Mean). So the measure with the smallest value is Mode? Wait, no, wait, maybe I made a mistake. Wait, let's check again. Wait, the data set: 70,75,80,80,82,85,88,90,91,99. Sum is 840, mean 84. Median: (82 + 85)/2 = 83.5. Mode is 80. So 80 < 83.5 < 84. So the mode has the smallest value. Wait, but the question is "which of the following is the measure of central tendency with the smallest value". The options are Mean, Median, Mode. So the answer is Mode? Wait, but let's confirm:

Wait, maybe I miscalculated. Let's re - check the sum:

70 + 75 = 145

145+80 = 225

225+80 = 305

305+82 = 387

387+85 = 472

472+88 = 560

560+90 = 650

650+91 = 741

741+99 = 840. Correct. Mean: 840/10 = 84. Correct.

Median: 10 data points, so average of 5th and 6th. 5th is 82, 6th is 85. (82 + 85)/2 = 83.5. Correct.

Mode: 80 appears twice, others once. So Mode is 80. So 80 is less than 83.5 and 84. So the measure with the smallest value is Mode. Wait, but the options are Mean, Median, Mode. So the answer is Mode? Wait, but let's check again. Wait, maybe I messed up the median. Wait, the ordered data: 70,75,80,80,82,85,88,90,91,99. The 5th term is 82, 6th is 85. So median is (82 + 85)/2 = 83.5. Correct. Mean is 84. Mode is 80. So 80 < 83.5 < 84. So the mode has the smallest value. So the answer is Mode. Wait, but the options are presented as radio buttons: Mean, Median, Mode. So the correct option is Mode? Wait, but let's confirm once more.

Wait, maybe I made a mistake in the mode. Wait, 80 appears twice, others once. So mode is 80. Correct. Mean 84, median 83.5, mode 80. So mode is the smallest. So the answer is Mode.

Answer:

Mode