In A Hearing Test Subjects Estimate Loudness In Decibels
1in A Hearing Test Subjects Estimate The Loudness In Decibels Of A
In a hearing test, subjects estimate the loudness (in decibels) of a sound, and the results are: 69, 67, 71, 72, 65, 75, 68, 68, 83, 73, 68. Calculate the measures of central tendency (Mean, median, mode) and the measures of dispersion (range, standard deviation, variance).
The data collected from the hearing test provides a foundation for analyzing central tendency and dispersion, critical for understanding the distribution and variability of loudness perceptions among subjects. Calculating the mean, median, and mode will give insight into the central point and common values, while the range, variance, and standard deviation offer a measure of spread and dispersion.
Paper For Above instruction
The analysis of loudness estimates from subjects in a hearing test involves different statistical measures that depict the data's central tendency and variability. These measures help quantify the typical perceived loudness and the consistency of the test results among various subjects. In this context, the mean, median, and mode serve as measures of central tendency, while the range, variance, and standard deviation provide the dispersion metrics.
Calculations of Measures of Central Tendency
First, the mean (average) is calculated by summing all data points and dividing by the total number of observations:
Mean = (69 + 67 + 71 + 72 + 65 + 75 + 68 + 68 + 83 + 73 + 68) / 11 = 790 / 11 ≈ 71.82 decibels.
The median is the middle value when the data set is ordered from smallest to largest. ordering the data: 65, 67, 68, 68, 68, 69, 71, 72, 73, 75, 83. Since there are 11 observations, the median is the 6th value, which is 69.
The mode indicates the most frequently occurring value. Here, 68 appears three times, more than any other value, making 68 the mode.
Calculations of Measures of Dispersion
The range is the difference between the maximum and minimum values:
Range = 83 - 65 = 18 decibels.
Next, the variance quantifies the average squared deviation from the mean. The calculations involve:
- Calculating each data point's deviation from the mean.
- Squaring each deviation.
- Calculating the average of these squared deviations.
Deviations squared:
- (69 - 71.82)^2 ≈ 8.07
- (67 - 71.82)^2 ≈ 23.21
- (71 - 71.82)^2 ≈ 0.67
- (72 - 71.82)^2 ≈ 0.03
- (65 - 71.82)^2 ≈ 47.59
- (75 - 71.82)^2 ≈ 10.02
- (68 - 71.82)^2 ≈ 14.58
- (68 - 71.82)^2 ≈ 14.58
- (83 - 71.82)^2 ≈ 124.74
- (73 - 71.82)^2 ≈ 1.39
- (68 - 71.82)^2 ≈ 14.58
Total sum of squared deviations: approximately 259.66. Variance = 259.66 / (n - 1) = 259.66 / 10 ≈ 25.97.
The standard deviation is the square root of the variance:
Standard deviation ≈ √25.97 ≈ 5.10 decibels.
Conclusion
In summary, the loudness estimates have a mean of approximately 71.82 dB, median of 69 dB, and mode of 68 dB. The data's spread is represented by a range of 18 dB, a variance of approximately 25.97, and a standard deviation of about 5.10 dB. These statistics provide a comprehensive understanding of perceived loudness levels in the sample and can inform hearing sensitivity analyses or auditory research.
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