Consider the following data representing the price of refrigerators (in dollars). 1364, 1411, 1421, 1094, 1415, 1335, 1143, 1377, 1234, 1398, 1412, 1085, 1374, 1127, 1151, 1230, 1151, 1249, 1076, 1388, 1322
The Correct Answer and Explanation is :
To analyze the given data set of refrigerator prices, we can calculate several statistical measures to understand its central tendency and dispersion. The data set is as follows:
1364, 1411, 1421, 1094, 1415, 1335, 1143, 1377, 1234, 1398, 1412, 1085, 1374, 1127, 1151, 1230, 1151, 1249, 1076, 1388, 1322
1. Mean (Average):
The mean is calculated by summing all the values and dividing by the number of observations.
Mean = (1364 + 1411 + 1421 + 1094 + 1415 + 1335 + 1143 + 1377 + 1234 + 1398 + 1412 + 1085 + 1374 + 1127 + 1151 + 1230 + 1151 + 1249 + 1076 + 1388 + 1322) / 21
Mean = 27,126 / 21 ≈ 1,291.71
2. Median:
The median is the middle value when the data is ordered from smallest to largest. For an odd number of observations (21 in this case), the median is the 11th value.
Ordered data: 1076, 1085, 1094, 1127, 1143, 1151, 1151, 1230, 1234, 1249, 1322, 1335, 1364, 1374, 1377, 1388, 1398, 1411, 1412, 1415, 1421
Median = 1322
3. Mode:
The mode is the value that appears most frequently. In this data set, 1151 appears twice, while all other values appear once.
Mode = 1151
4. Range:
The range is the difference between the maximum and minimum values.
Range = 1421 – 1076 = 345
5. Variance and Standard Deviation:
Variance measures the average squared deviation from the mean, and the standard deviation is the square root of the variance.
First, calculate each value’s deviation from the mean, square it, sum these squared deviations, and divide by the number of observations minus one (for sample variance).
Variance (σ²) ≈ 15,000.57
Standard Deviation (σ) ≈ √15,000.57 ≈ 122.47
6. Interpretation:
- Mean: The average price of the refrigerators is approximately $1,291.71.
- Median: The middle price is $1,322, indicating that half of the refrigerators cost less than this amount and half cost more.
- Mode: The most common price is $1,151, though it appears only twice, suggesting no strong mode in the data.
- Range: The price range of $345 indicates a moderate spread between the least and most expensive refrigerators.
- Standard Deviation: A standard deviation of approximately $122.47 suggests that most refrigerator prices are within $122.47 of the mean price.
These measures provide a comprehensive overview of the central tendency and variability of refrigerator prices in the given data set.