Machine learning algorithms are increasingly used in making people’s life decisions across a range of different areas such as loan applications, university admissions, insurance pricing and criminal justice sentencing. If the historical data used to train the algorithm is biased against certain demographic groups (e.g. black people or women), the predictive results of the algorithms will too. From both regulation and ethical perspectives, we need to reduce discrimination and improve group fairness which concentrates on equalizing the outcomes across distinct groups. However, there are cases where the outcomes are unfair from an individual’s point of view when group fairness is satisfied. Individual fairness states that similar individuals should be treated similarly. It is also an important concept of fairness and needs to be considered carefully while we improve group fairness, but it has not yet received much attention in the literature. Most of the existing fairness algorithms concentrates on achieving group fairness disregarding individual fairness. It is important to explore the relationship between group fairness and individual fairness, specifically, in what cases the individual fairness would be affected when we improve group fairness. We show by practical results from real data sets that, after removing the sensitive attributes, there generally exists a trade off between group fairness and individual fairness. Moreover, we use experimental results from simulated data sets to show that satisfying group fairness decreases the level of individual fairness when the Wasserstein distance (which is a measure of the distance between two distributions) between the attribute distributions of two groups is large. By adjusting the parameters of the simulated distributions, we show that, if a large Wasserstein distance is caused by a large mean difference rather than a large variance difference, individual fairness is more likely to be affected when group fairness is satisfied. Furthermore, we not only tweak the existing reweighing algorithm to obtain more flexible performance on individual fairness and group fairness, but also construct a new algorithm to achieve fairness. This approach reduces the mean difference in attribute values between different groups so that the association between the sensitive attribute and non-sensitive attributes is decreased. This method can be used to achieve fairness among more than two demographic groups and solve fairness problems in multi-classification or regression scenarios. We assess the performance of this method in terms of both group fairness and individual fairness and the results show that our method outperforms two existing fairness algorithms: reweighing and reject option based classification.